Step |
Hyp |
Ref |
Expression |
1 |
|
supgtoreq.1 |
⊢ ( 𝜑 → 𝑅 Or 𝐴 ) |
2 |
|
supgtoreq.2 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
3 |
|
supgtoreq.3 |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
4 |
|
supgtoreq.4 |
⊢ ( 𝜑 → 𝐶 ∈ 𝐵 ) |
5 |
|
supgtoreq.5 |
⊢ ( 𝜑 → 𝑆 = sup ( 𝐵 , 𝐴 , 𝑅 ) ) |
6 |
4
|
ne0d |
⊢ ( 𝜑 → 𝐵 ≠ ∅ ) |
7 |
|
fisup2g |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴 ) ) → ∃ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |
8 |
1 3 6 2 7
|
syl13anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |
9 |
|
ssrexv |
⊢ ( 𝐵 ⊆ 𝐴 → ( ∃ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) ) |
10 |
2 8 9
|
sylc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) ) |
11 |
1 10
|
supub |
⊢ ( 𝜑 → ( 𝐶 ∈ 𝐵 → ¬ sup ( 𝐵 , 𝐴 , 𝑅 ) 𝑅 𝐶 ) ) |
12 |
4 11
|
mpd |
⊢ ( 𝜑 → ¬ sup ( 𝐵 , 𝐴 , 𝑅 ) 𝑅 𝐶 ) |
13 |
5 12
|
eqnbrtrd |
⊢ ( 𝜑 → ¬ 𝑆 𝑅 𝐶 ) |
14 |
|
fisupcl |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴 ) ) → sup ( 𝐵 , 𝐴 , 𝑅 ) ∈ 𝐵 ) |
15 |
1 3 6 2 14
|
syl13anc |
⊢ ( 𝜑 → sup ( 𝐵 , 𝐴 , 𝑅 ) ∈ 𝐵 ) |
16 |
2 15
|
sseldd |
⊢ ( 𝜑 → sup ( 𝐵 , 𝐴 , 𝑅 ) ∈ 𝐴 ) |
17 |
5 16
|
eqeltrd |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
18 |
2 4
|
sseldd |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
19 |
|
sotric |
⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝑆 𝑅 𝐶 ↔ ¬ ( 𝑆 = 𝐶 ∨ 𝐶 𝑅 𝑆 ) ) ) |
20 |
1 17 18 19
|
syl12anc |
⊢ ( 𝜑 → ( 𝑆 𝑅 𝐶 ↔ ¬ ( 𝑆 = 𝐶 ∨ 𝐶 𝑅 𝑆 ) ) ) |
21 |
|
orcom |
⊢ ( ( 𝑆 = 𝐶 ∨ 𝐶 𝑅 𝑆 ) ↔ ( 𝐶 𝑅 𝑆 ∨ 𝑆 = 𝐶 ) ) |
22 |
|
eqcom |
⊢ ( 𝑆 = 𝐶 ↔ 𝐶 = 𝑆 ) |
23 |
22
|
orbi2i |
⊢ ( ( 𝐶 𝑅 𝑆 ∨ 𝑆 = 𝐶 ) ↔ ( 𝐶 𝑅 𝑆 ∨ 𝐶 = 𝑆 ) ) |
24 |
21 23
|
bitri |
⊢ ( ( 𝑆 = 𝐶 ∨ 𝐶 𝑅 𝑆 ) ↔ ( 𝐶 𝑅 𝑆 ∨ 𝐶 = 𝑆 ) ) |
25 |
24
|
notbii |
⊢ ( ¬ ( 𝑆 = 𝐶 ∨ 𝐶 𝑅 𝑆 ) ↔ ¬ ( 𝐶 𝑅 𝑆 ∨ 𝐶 = 𝑆 ) ) |
26 |
20 25
|
bitr2di |
⊢ ( 𝜑 → ( ¬ ( 𝐶 𝑅 𝑆 ∨ 𝐶 = 𝑆 ) ↔ 𝑆 𝑅 𝐶 ) ) |
27 |
13 26
|
mtbird |
⊢ ( 𝜑 → ¬ ¬ ( 𝐶 𝑅 𝑆 ∨ 𝐶 = 𝑆 ) ) |
28 |
27
|
notnotrd |
⊢ ( 𝜑 → ( 𝐶 𝑅 𝑆 ∨ 𝐶 = 𝑆 ) ) |