| Step |
Hyp |
Ref |
Expression |
| 1 |
|
xpord3.1 |
⊢ 𝑈 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑦 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ ( ( ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ( 1st ‘ ( 1st ‘ 𝑦 ) ) ∨ ( 1st ‘ ( 1st ‘ 𝑥 ) ) = ( 1st ‘ ( 1st ‘ 𝑦 ) ) ) ∧ ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑆 ( 2nd ‘ ( 1st ‘ 𝑦 ) ) ∨ ( 2nd ‘ ( 1st ‘ 𝑥 ) ) = ( 2nd ‘ ( 1st ‘ 𝑦 ) ) ) ∧ ( ( 2nd ‘ 𝑥 ) 𝑇 ( 2nd ‘ 𝑦 ) ∨ ( 2nd ‘ 𝑥 ) = ( 2nd ‘ 𝑦 ) ) ) ∧ 𝑥 ≠ 𝑦 ) ) } |
| 2 |
|
poxp3.1 |
⊢ ( 𝜑 → 𝑅 Po 𝐴 ) |
| 3 |
|
poxp3.2 |
⊢ ( 𝜑 → 𝑆 Po 𝐵 ) |
| 4 |
|
poxp3.3 |
⊢ ( 𝜑 → 𝑇 Po 𝐶 ) |
| 5 |
|
elxpxp |
⊢ ( 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ↔ ∃ 𝑑 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ) |
| 6 |
|
neirr |
⊢ ¬ 𝑑 ≠ 𝑑 |
| 7 |
|
neirr |
⊢ ¬ 𝑒 ≠ 𝑒 |
| 8 |
|
neirr |
⊢ ¬ 𝑓 ≠ 𝑓 |
| 9 |
6 7 8
|
3pm3.2ni |
⊢ ¬ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) |
| 10 |
9
|
intnan |
⊢ ¬ ( ( ( 𝑑 𝑅 𝑑 ∨ 𝑑 = 𝑑 ) ∧ ( 𝑒 𝑆 𝑒 ∨ 𝑒 = 𝑒 ) ∧ ( 𝑓 𝑇 𝑓 ∨ 𝑓 = 𝑓 ) ) ∧ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) ) |
| 11 |
|
simp3 |
⊢ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑑 ∨ 𝑑 = 𝑑 ) ∧ ( 𝑒 𝑆 𝑒 ∨ 𝑒 = 𝑒 ) ∧ ( 𝑓 𝑇 𝑓 ∨ 𝑓 = 𝑓 ) ) ∧ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) ) ) → ( ( ( 𝑑 𝑅 𝑑 ∨ 𝑑 = 𝑑 ) ∧ ( 𝑒 𝑆 𝑒 ∨ 𝑒 = 𝑒 ) ∧ ( 𝑓 𝑇 𝑓 ∨ 𝑓 = 𝑓 ) ) ∧ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) ) ) |
| 12 |
10 11
|
mto |
⊢ ¬ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑑 ∨ 𝑑 = 𝑑 ) ∧ ( 𝑒 𝑆 𝑒 ∨ 𝑒 = 𝑒 ) ∧ ( 𝑓 𝑇 𝑓 ∨ 𝑓 = 𝑓 ) ) ∧ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) ) ) |
| 13 |
|
breq12 |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ) → ( 𝑎 𝑈 𝑎 ↔ ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ) ) |
| 14 |
13
|
anidms |
⊢ ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ → ( 𝑎 𝑈 𝑎 ↔ ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ) ) |
| 15 |
1
|
xpord3lem |
⊢ ( ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ↔ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑑 ∨ 𝑑 = 𝑑 ) ∧ ( 𝑒 𝑆 𝑒 ∨ 𝑒 = 𝑒 ) ∧ ( 𝑓 𝑇 𝑓 ∨ 𝑓 = 𝑓 ) ) ∧ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) ) ) ) |
| 16 |
14 15
|
bitrdi |
⊢ ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ → ( 𝑎 𝑈 𝑎 ↔ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑑 ∨ 𝑑 = 𝑑 ) ∧ ( 𝑒 𝑆 𝑒 ∨ 𝑒 = 𝑒 ) ∧ ( 𝑓 𝑇 𝑓 ∨ 𝑓 = 𝑓 ) ) ∧ ( 𝑑 ≠ 𝑑 ∨ 𝑒 ≠ 𝑒 ∨ 𝑓 ≠ 𝑓 ) ) ) ) ) |
| 17 |
12 16
|
mtbiri |
⊢ ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ → ¬ 𝑎 𝑈 𝑎 ) |
| 18 |
17
|
rexlimivw |
⊢ ( ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ → ¬ 𝑎 𝑈 𝑎 ) |
| 19 |
18
|
a1i |
⊢ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ) → ( ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ → ¬ 𝑎 𝑈 𝑎 ) ) |
| 20 |
19
|
rexlimivv |
⊢ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ → ¬ 𝑎 𝑈 𝑎 ) |
| 21 |
5 20
|
sylbi |
⊢ ( 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) → ¬ 𝑎 𝑈 𝑎 ) |
| 22 |
21
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ) → ¬ 𝑎 𝑈 𝑎 ) |
| 23 |
|
elxpxp |
⊢ ( 𝑏 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ↔ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ) |
| 24 |
|
elxpxp |
⊢ ( 𝑐 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ↔ ∃ 𝑗 ∈ 𝐴 ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) |
| 25 |
5 23 24
|
3anbi123i |
⊢ ( ( 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑏 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑐 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ) ↔ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑗 ∈ 𝐴 ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 26 |
|
3reeanv |
⊢ ( ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ( ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 27 |
26
|
rexbii |
⊢ ( ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ∃ 𝑘 ∈ 𝐵 ( ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 28 |
27
|
2rexbii |
⊢ ( ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ( ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 29 |
|
3reeanv |
⊢ ( ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ( ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ( ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 30 |
28 29
|
bitri |
⊢ ( ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ( ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 31 |
30
|
rexbii |
⊢ ( ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ∃ 𝑗 ∈ 𝐴 ( ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 32 |
31
|
2rexbii |
⊢ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑔 ∈ 𝐴 ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ∃ 𝑑 ∈ 𝐴 ∃ 𝑔 ∈ 𝐴 ∃ 𝑗 ∈ 𝐴 ( ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 33 |
|
3reeanv |
⊢ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑔 ∈ 𝐴 ∃ 𝑗 ∈ 𝐴 ( ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑗 ∈ 𝐴 ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 34 |
32 33
|
bitri |
⊢ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑔 ∈ 𝐴 ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ↔ ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ ∃ 𝑔 ∈ 𝐴 ∃ ℎ ∈ 𝐵 ∃ 𝑖 ∈ 𝐶 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ ∃ 𝑗 ∈ 𝐴 ∃ 𝑘 ∈ 𝐵 ∃ 𝑙 ∈ 𝐶 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 35 |
25 34
|
bitr4i |
⊢ ( ( 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑏 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑐 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ) ↔ ∃ 𝑑 ∈ 𝐴 ∃ 𝑔 ∈ 𝐴 ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 36 |
|
simprl1 |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ) |
| 37 |
|
simprr2 |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ) |
| 38 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑅 Po 𝐴 ) |
| 39 |
|
simpl11 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑑 ∈ 𝐴 ) |
| 40 |
39
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑑 ∈ 𝐴 ) |
| 41 |
|
simpr11 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑔 ∈ 𝐴 ) |
| 42 |
41
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑔 ∈ 𝐴 ) |
| 43 |
|
simpr21 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑗 ∈ 𝐴 ) |
| 44 |
43
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑗 ∈ 𝐴 ) |
| 45 |
|
potr |
⊢ ( ( 𝑅 Po 𝐴 ∧ ( 𝑑 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ∧ 𝑗 ∈ 𝐴 ) ) → ( ( 𝑑 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) → 𝑑 𝑅 𝑗 ) ) |
| 46 |
38 40 42 44 45
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑑 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) → 𝑑 𝑅 𝑗 ) ) |
| 47 |
|
orc |
⊢ ( 𝑑 𝑅 𝑗 → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) |
| 48 |
46 47
|
syl6 |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑑 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) |
| 49 |
48
|
expd |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 𝑅 𝑔 → ( 𝑔 𝑅 𝑗 → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) ) |
| 50 |
|
breq1 |
⊢ ( 𝑑 = 𝑔 → ( 𝑑 𝑅 𝑗 ↔ 𝑔 𝑅 𝑗 ) ) |
| 51 |
50 47
|
syl6bir |
⊢ ( 𝑑 = 𝑔 → ( 𝑔 𝑅 𝑗 → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) |
| 52 |
51
|
a1i |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 = 𝑔 → ( 𝑔 𝑅 𝑗 → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) ) |
| 53 |
|
simp3l1 |
⊢ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) → ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ) |
| 54 |
53
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ) |
| 55 |
49 52 54
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑔 𝑅 𝑗 → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) |
| 56 |
|
breq2 |
⊢ ( 𝑔 = 𝑗 → ( 𝑑 𝑅 𝑔 ↔ 𝑑 𝑅 𝑗 ) ) |
| 57 |
|
equequ2 |
⊢ ( 𝑔 = 𝑗 → ( 𝑑 = 𝑔 ↔ 𝑑 = 𝑗 ) ) |
| 58 |
56 57
|
orbi12d |
⊢ ( 𝑔 = 𝑗 → ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ↔ ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) |
| 59 |
54 58
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑔 = 𝑗 → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) ) |
| 60 |
|
simp3l1 |
⊢ ( ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) → ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ) |
| 61 |
60
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ) |
| 62 |
55 59 61
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ) |
| 63 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑆 Po 𝐵 ) |
| 64 |
|
simpl12 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑒 ∈ 𝐵 ) |
| 65 |
64
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑒 ∈ 𝐵 ) |
| 66 |
|
simpr12 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → ℎ ∈ 𝐵 ) |
| 67 |
66
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ℎ ∈ 𝐵 ) |
| 68 |
|
simpr22 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑘 ∈ 𝐵 ) |
| 69 |
68
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑘 ∈ 𝐵 ) |
| 70 |
|
potr |
⊢ ( ( 𝑆 Po 𝐵 ∧ ( 𝑒 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ∧ 𝑘 ∈ 𝐵 ) ) → ( ( 𝑒 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) → 𝑒 𝑆 𝑘 ) ) |
| 71 |
63 65 67 69 70
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑒 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) → 𝑒 𝑆 𝑘 ) ) |
| 72 |
|
orc |
⊢ ( 𝑒 𝑆 𝑘 → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) |
| 73 |
71 72
|
syl6 |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑒 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) |
| 74 |
73
|
expd |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑒 𝑆 ℎ → ( ℎ 𝑆 𝑘 → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) ) |
| 75 |
|
breq1 |
⊢ ( 𝑒 = ℎ → ( 𝑒 𝑆 𝑘 ↔ ℎ 𝑆 𝑘 ) ) |
| 76 |
75 72
|
syl6bir |
⊢ ( 𝑒 = ℎ → ( ℎ 𝑆 𝑘 → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) |
| 77 |
76
|
a1i |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑒 = ℎ → ( ℎ 𝑆 𝑘 → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) ) |
| 78 |
|
simp3l2 |
⊢ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) → ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ) |
| 79 |
78
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ) |
| 80 |
74 77 79
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ℎ 𝑆 𝑘 → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) |
| 81 |
|
breq2 |
⊢ ( ℎ = 𝑘 → ( 𝑒 𝑆 ℎ ↔ 𝑒 𝑆 𝑘 ) ) |
| 82 |
|
equequ2 |
⊢ ( ℎ = 𝑘 → ( 𝑒 = ℎ ↔ 𝑒 = 𝑘 ) ) |
| 83 |
81 82
|
orbi12d |
⊢ ( ℎ = 𝑘 → ( ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ↔ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) |
| 84 |
79 83
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ℎ = 𝑘 → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) ) |
| 85 |
|
simp3l2 |
⊢ ( ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) → ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ) |
| 86 |
85
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ) |
| 87 |
80 84 86
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ) |
| 88 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑇 Po 𝐶 ) |
| 89 |
|
simpl13 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑓 ∈ 𝐶 ) |
| 90 |
89
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑓 ∈ 𝐶 ) |
| 91 |
|
simpr13 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑖 ∈ 𝐶 ) |
| 92 |
91
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑖 ∈ 𝐶 ) |
| 93 |
|
simpr23 |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → 𝑙 ∈ 𝐶 ) |
| 94 |
93
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → 𝑙 ∈ 𝐶 ) |
| 95 |
|
potr |
⊢ ( ( 𝑇 Po 𝐶 ∧ ( 𝑓 ∈ 𝐶 ∧ 𝑖 ∈ 𝐶 ∧ 𝑙 ∈ 𝐶 ) ) → ( ( 𝑓 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) → 𝑓 𝑇 𝑙 ) ) |
| 96 |
88 90 92 94 95
|
syl13anc |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑓 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) → 𝑓 𝑇 𝑙 ) ) |
| 97 |
|
orc |
⊢ ( 𝑓 𝑇 𝑙 → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) |
| 98 |
96 97
|
syl6 |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑓 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) |
| 99 |
98
|
expd |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑓 𝑇 𝑖 → ( 𝑖 𝑇 𝑙 → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) ) |
| 100 |
|
breq1 |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 𝑇 𝑙 ↔ 𝑖 𝑇 𝑙 ) ) |
| 101 |
100 97
|
syl6bir |
⊢ ( 𝑓 = 𝑖 → ( 𝑖 𝑇 𝑙 → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) |
| 102 |
101
|
a1i |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑓 = 𝑖 → ( 𝑖 𝑇 𝑙 → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) ) |
| 103 |
|
simp3l3 |
⊢ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) → ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) |
| 104 |
103
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) |
| 105 |
99 102 104
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑖 𝑇 𝑙 → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) |
| 106 |
|
breq2 |
⊢ ( 𝑖 = 𝑙 → ( 𝑓 𝑇 𝑖 ↔ 𝑓 𝑇 𝑙 ) ) |
| 107 |
|
equequ2 |
⊢ ( 𝑖 = 𝑙 → ( 𝑓 = 𝑖 ↔ 𝑓 = 𝑙 ) ) |
| 108 |
106 107
|
orbi12d |
⊢ ( 𝑖 = 𝑙 → ( ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ↔ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) |
| 109 |
104 108
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑖 = 𝑙 → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) |
| 110 |
|
simp3l3 |
⊢ ( ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) → ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) |
| 111 |
110
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) |
| 112 |
105 109 111
|
mpjaod |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) |
| 113 |
62 87 112
|
3jca |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ) |
| 114 |
|
simpr3r |
⊢ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) |
| 115 |
114
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) |
| 116 |
|
df-ne |
⊢ ( 𝑔 ≠ 𝑗 ↔ ¬ 𝑔 = 𝑗 ) |
| 117 |
|
df-ne |
⊢ ( ℎ ≠ 𝑘 ↔ ¬ ℎ = 𝑘 ) |
| 118 |
|
df-ne |
⊢ ( 𝑖 ≠ 𝑙 ↔ ¬ 𝑖 = 𝑙 ) |
| 119 |
116 117 118
|
3orbi123i |
⊢ ( ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ↔ ( ¬ 𝑔 = 𝑗 ∨ ¬ ℎ = 𝑘 ∨ ¬ 𝑖 = 𝑙 ) ) |
| 120 |
|
3ianor |
⊢ ( ¬ ( 𝑔 = 𝑗 ∧ ℎ = 𝑘 ∧ 𝑖 = 𝑙 ) ↔ ( ¬ 𝑔 = 𝑗 ∨ ¬ ℎ = 𝑘 ∨ ¬ 𝑖 = 𝑙 ) ) |
| 121 |
119 120
|
bitr4i |
⊢ ( ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ↔ ¬ ( 𝑔 = 𝑗 ∧ ℎ = 𝑘 ∧ 𝑖 = 𝑙 ) ) |
| 122 |
115 121
|
sylib |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ¬ ( 𝑔 = 𝑗 ∧ ℎ = 𝑘 ∧ 𝑖 = 𝑙 ) ) |
| 123 |
|
df-ne |
⊢ ( 𝑑 ≠ 𝑗 ↔ ¬ 𝑑 = 𝑗 ) |
| 124 |
|
df-ne |
⊢ ( 𝑒 ≠ 𝑘 ↔ ¬ 𝑒 = 𝑘 ) |
| 125 |
|
df-ne |
⊢ ( 𝑓 ≠ 𝑙 ↔ ¬ 𝑓 = 𝑙 ) |
| 126 |
123 124 125
|
3orbi123i |
⊢ ( ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ↔ ( ¬ 𝑑 = 𝑗 ∨ ¬ 𝑒 = 𝑘 ∨ ¬ 𝑓 = 𝑙 ) ) |
| 127 |
|
3ianor |
⊢ ( ¬ ( 𝑑 = 𝑗 ∧ 𝑒 = 𝑘 ∧ 𝑓 = 𝑙 ) ↔ ( ¬ 𝑑 = 𝑗 ∨ ¬ 𝑒 = 𝑘 ∨ ¬ 𝑓 = 𝑙 ) ) |
| 128 |
126 127
|
bitr4i |
⊢ ( ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ↔ ¬ ( 𝑑 = 𝑗 ∧ 𝑒 = 𝑘 ∧ 𝑓 = 𝑙 ) ) |
| 129 |
128
|
con2bii |
⊢ ( ( 𝑑 = 𝑗 ∧ 𝑒 = 𝑘 ∧ 𝑓 = 𝑙 ) ↔ ¬ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) |
| 130 |
|
breq1 |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 𝑅 𝑔 ↔ 𝑗 𝑅 𝑔 ) ) |
| 131 |
|
equequ1 |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 = 𝑔 ↔ 𝑗 = 𝑔 ) ) |
| 132 |
|
equcom |
⊢ ( 𝑗 = 𝑔 ↔ 𝑔 = 𝑗 ) |
| 133 |
131 132
|
bitrdi |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 = 𝑔 ↔ 𝑔 = 𝑗 ) ) |
| 134 |
130 133
|
orbi12d |
⊢ ( 𝑑 = 𝑗 → ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ↔ ( 𝑗 𝑅 𝑔 ∨ 𝑔 = 𝑗 ) ) ) |
| 135 |
54 134
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 = 𝑗 → ( 𝑗 𝑅 𝑔 ∨ 𝑔 = 𝑗 ) ) ) |
| 136 |
135
|
imp |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑑 = 𝑗 ) → ( 𝑗 𝑅 𝑔 ∨ 𝑔 = 𝑗 ) ) |
| 137 |
61
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑑 = 𝑗 ) → ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ) |
| 138 |
|
ordir |
⊢ ( ( ( 𝑗 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) ∨ 𝑔 = 𝑗 ) ↔ ( ( 𝑗 𝑅 𝑔 ∨ 𝑔 = 𝑗 ) ∧ ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ) ) |
| 139 |
136 137 138
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑑 = 𝑗 ) → ( ( 𝑗 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) ∨ 𝑔 = 𝑗 ) ) |
| 140 |
|
po2nr |
⊢ ( ( 𝑅 Po 𝐴 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) ) → ¬ ( 𝑗 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) ) |
| 141 |
38 44 42 140
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ¬ ( 𝑗 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) ) |
| 142 |
141
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑑 = 𝑗 ) → ¬ ( 𝑗 𝑅 𝑔 ∧ 𝑔 𝑅 𝑗 ) ) |
| 143 |
139 142
|
orcnd |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑑 = 𝑗 ) → 𝑔 = 𝑗 ) |
| 144 |
143
|
ex |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 = 𝑗 → 𝑔 = 𝑗 ) ) |
| 145 |
|
breq1 |
⊢ ( 𝑒 = 𝑘 → ( 𝑒 𝑆 ℎ ↔ 𝑘 𝑆 ℎ ) ) |
| 146 |
|
equequ1 |
⊢ ( 𝑒 = 𝑘 → ( 𝑒 = ℎ ↔ 𝑘 = ℎ ) ) |
| 147 |
|
equcom |
⊢ ( 𝑘 = ℎ ↔ ℎ = 𝑘 ) |
| 148 |
146 147
|
bitrdi |
⊢ ( 𝑒 = 𝑘 → ( 𝑒 = ℎ ↔ ℎ = 𝑘 ) ) |
| 149 |
145 148
|
orbi12d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ↔ ( 𝑘 𝑆 ℎ ∨ ℎ = 𝑘 ) ) ) |
| 150 |
79 149
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑒 = 𝑘 → ( 𝑘 𝑆 ℎ ∨ ℎ = 𝑘 ) ) ) |
| 151 |
150
|
imp |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑒 = 𝑘 ) → ( 𝑘 𝑆 ℎ ∨ ℎ = 𝑘 ) ) |
| 152 |
86
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑒 = 𝑘 ) → ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ) |
| 153 |
|
ordir |
⊢ ( ( ( 𝑘 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) ∨ ℎ = 𝑘 ) ↔ ( ( 𝑘 𝑆 ℎ ∨ ℎ = 𝑘 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ) ) |
| 154 |
151 152 153
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑒 = 𝑘 ) → ( ( 𝑘 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) ∨ ℎ = 𝑘 ) ) |
| 155 |
|
po2nr |
⊢ ( ( 𝑆 Po 𝐵 ∧ ( 𝑘 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) ) → ¬ ( 𝑘 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) ) |
| 156 |
63 69 67 155
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ¬ ( 𝑘 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) ) |
| 157 |
156
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑒 = 𝑘 ) → ¬ ( 𝑘 𝑆 ℎ ∧ ℎ 𝑆 𝑘 ) ) |
| 158 |
154 157
|
orcnd |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑒 = 𝑘 ) → ℎ = 𝑘 ) |
| 159 |
158
|
ex |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑒 = 𝑘 → ℎ = 𝑘 ) ) |
| 160 |
|
breq1 |
⊢ ( 𝑓 = 𝑙 → ( 𝑓 𝑇 𝑖 ↔ 𝑙 𝑇 𝑖 ) ) |
| 161 |
|
equequ1 |
⊢ ( 𝑓 = 𝑙 → ( 𝑓 = 𝑖 ↔ 𝑙 = 𝑖 ) ) |
| 162 |
160 161
|
orbi12d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ↔ ( 𝑙 𝑇 𝑖 ∨ 𝑙 = 𝑖 ) ) ) |
| 163 |
104 162
|
syl5ibcom |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑓 = 𝑙 → ( 𝑙 𝑇 𝑖 ∨ 𝑙 = 𝑖 ) ) ) |
| 164 |
163
|
imp |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑓 = 𝑙 ) → ( 𝑙 𝑇 𝑖 ∨ 𝑙 = 𝑖 ) ) |
| 165 |
|
equcom |
⊢ ( 𝑙 = 𝑖 ↔ 𝑖 = 𝑙 ) |
| 166 |
165
|
orbi2i |
⊢ ( ( 𝑙 𝑇 𝑖 ∨ 𝑙 = 𝑖 ) ↔ ( 𝑙 𝑇 𝑖 ∨ 𝑖 = 𝑙 ) ) |
| 167 |
164 166
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑓 = 𝑙 ) → ( 𝑙 𝑇 𝑖 ∨ 𝑖 = 𝑙 ) ) |
| 168 |
111
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑓 = 𝑙 ) → ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) |
| 169 |
|
ordir |
⊢ ( ( ( 𝑙 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) ∨ 𝑖 = 𝑙 ) ↔ ( ( 𝑙 𝑇 𝑖 ∨ 𝑖 = 𝑙 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ) |
| 170 |
167 168 169
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑓 = 𝑙 ) → ( ( 𝑙 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) ∨ 𝑖 = 𝑙 ) ) |
| 171 |
|
po2nr |
⊢ ( ( 𝑇 Po 𝐶 ∧ ( 𝑙 ∈ 𝐶 ∧ 𝑖 ∈ 𝐶 ) ) → ¬ ( 𝑙 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) ) |
| 172 |
88 94 92 171
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ¬ ( 𝑙 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) ) |
| 173 |
172
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑓 = 𝑙 ) → ¬ ( 𝑙 𝑇 𝑖 ∧ 𝑖 𝑇 𝑙 ) ) |
| 174 |
170 173
|
orcnd |
⊢ ( ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ∧ 𝑓 = 𝑙 ) → 𝑖 = 𝑙 ) |
| 175 |
174
|
ex |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑓 = 𝑙 → 𝑖 = 𝑙 ) ) |
| 176 |
144 159 175
|
3anim123d |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑑 = 𝑗 ∧ 𝑒 = 𝑘 ∧ 𝑓 = 𝑙 ) → ( 𝑔 = 𝑗 ∧ ℎ = 𝑘 ∧ 𝑖 = 𝑙 ) ) ) |
| 177 |
129 176
|
syl5bir |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ¬ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) → ( 𝑔 = 𝑗 ∧ ℎ = 𝑘 ∧ 𝑖 = 𝑙 ) ) ) |
| 178 |
122 177
|
mt3d |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) |
| 179 |
113 178
|
jca |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ∧ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) ) |
| 180 |
36 37 179
|
3jca |
⊢ ( ( 𝜑 ∧ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) → ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ∧ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) ) ) |
| 181 |
180
|
ex |
⊢ ( 𝜑 → ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ∧ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) ) ) ) |
| 182 |
|
breq12 |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ) → ( 𝑎 𝑈 𝑏 ↔ ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ) ) |
| 183 |
182
|
3adant3 |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑎 𝑈 𝑏 ↔ ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ) ) |
| 184 |
1
|
xpord3lem |
⊢ ( ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ↔ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ) |
| 185 |
183 184
|
bitrdi |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑎 𝑈 𝑏 ↔ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ) ) |
| 186 |
|
breq12 |
⊢ ( ( 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑏 𝑈 𝑐 ↔ ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ 𝑈 ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 187 |
186
|
3adant1 |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑏 𝑈 𝑐 ↔ ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ 𝑈 ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 188 |
1
|
xpord3lem |
⊢ ( ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ 𝑈 ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ↔ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) |
| 189 |
187 188
|
bitrdi |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑏 𝑈 𝑐 ↔ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) |
| 190 |
185 189
|
anbi12d |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) ↔ ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) ) ) |
| 191 |
|
breq12 |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑎 𝑈 𝑐 ↔ ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 192 |
191
|
3adant2 |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑎 𝑈 𝑐 ↔ ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) ) |
| 193 |
1
|
xpord3lem |
⊢ ( ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ 𝑈 ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ↔ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ∧ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) ) ) |
| 194 |
192 193
|
bitrdi |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( 𝑎 𝑈 𝑐 ↔ ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ∧ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) ) ) ) |
| 195 |
190 194
|
imbi12d |
⊢ ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ↔ ( ( ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑔 ∨ 𝑑 = 𝑔 ) ∧ ( 𝑒 𝑆 ℎ ∨ 𝑒 = ℎ ) ∧ ( 𝑓 𝑇 𝑖 ∨ 𝑓 = 𝑖 ) ) ∧ ( 𝑑 ≠ 𝑔 ∨ 𝑒 ≠ ℎ ∨ 𝑓 ≠ 𝑖 ) ) ) ∧ ( ( 𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ 𝑖 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑔 𝑅 𝑗 ∨ 𝑔 = 𝑗 ) ∧ ( ℎ 𝑆 𝑘 ∨ ℎ = 𝑘 ) ∧ ( 𝑖 𝑇 𝑙 ∨ 𝑖 = 𝑙 ) ) ∧ ( 𝑔 ≠ 𝑗 ∨ ℎ ≠ 𝑘 ∨ 𝑖 ≠ 𝑙 ) ) ) ) → ( ( 𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶 ) ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ∧ 𝑙 ∈ 𝐶 ) ∧ ( ( ( 𝑑 𝑅 𝑗 ∨ 𝑑 = 𝑗 ) ∧ ( 𝑒 𝑆 𝑘 ∨ 𝑒 = 𝑘 ) ∧ ( 𝑓 𝑇 𝑙 ∨ 𝑓 = 𝑙 ) ) ∧ ( 𝑑 ≠ 𝑗 ∨ 𝑒 ≠ 𝑘 ∨ 𝑓 ≠ 𝑙 ) ) ) ) ) ) |
| 196 |
181 195
|
syl5ibrcom |
⊢ ( 𝜑 → ( ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 197 |
196
|
rexlimdvw |
⊢ ( 𝜑 → ( ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 198 |
197
|
a1d |
⊢ ( 𝜑 → ( ( 𝑓 ∈ 𝐶 ∧ 𝑖 ∈ 𝐶 ) → ( ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) ) |
| 199 |
198
|
rexlimdvv |
⊢ ( 𝜑 → ( ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 200 |
199
|
rexlimdvw |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 201 |
200
|
a1d |
⊢ ( 𝜑 → ( ( 𝑒 ∈ 𝐵 ∧ ℎ ∈ 𝐵 ) → ( ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) ) |
| 202 |
201
|
rexlimdvv |
⊢ ( 𝜑 → ( ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 203 |
202
|
rexlimdvw |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 204 |
203
|
a1d |
⊢ ( 𝜑 → ( ( 𝑑 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴 ) → ( ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) ) |
| 205 |
204
|
rexlimdvv |
⊢ ( 𝜑 → ( ∃ 𝑑 ∈ 𝐴 ∃ 𝑔 ∈ 𝐴 ∃ 𝑗 ∈ 𝐴 ∃ 𝑒 ∈ 𝐵 ∃ ℎ ∈ 𝐵 ∃ 𝑘 ∈ 𝐵 ∃ 𝑓 ∈ 𝐶 ∃ 𝑖 ∈ 𝐶 ∃ 𝑙 ∈ 𝐶 ( 𝑎 = ⟨ ⟨ 𝑑 , 𝑒 ⟩ , 𝑓 ⟩ ∧ 𝑏 = ⟨ ⟨ 𝑔 , ℎ ⟩ , 𝑖 ⟩ ∧ 𝑐 = ⟨ ⟨ 𝑗 , 𝑘 ⟩ , 𝑙 ⟩ ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 206 |
35 205
|
syl5bi |
⊢ ( 𝜑 → ( ( 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑏 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑐 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) ) |
| 207 |
206
|
imp |
⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑏 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ∧ 𝑐 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) ) ) → ( ( 𝑎 𝑈 𝑏 ∧ 𝑏 𝑈 𝑐 ) → 𝑎 𝑈 𝑐 ) ) |
| 208 |
22 207
|
ispod |
⊢ ( 𝜑 → 𝑈 Po ( ( 𝐴 × 𝐵 ) × 𝐶 ) ) |