Step |
Hyp |
Ref |
Expression |
1 |
|
oaordi |
⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ) ) |
2 |
1
|
3adant1 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ) ) |
3 |
|
oveq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝐶 +o 𝐴 ) = ( 𝐶 +o 𝐵 ) ) |
4 |
3
|
a1i |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 = 𝐵 → ( 𝐶 +o 𝐴 ) = ( 𝐶 +o 𝐵 ) ) ) |
5 |
|
oaordi |
⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 ∈ 𝐴 → ( 𝐶 +o 𝐵 ) ∈ ( 𝐶 +o 𝐴 ) ) ) |
6 |
5
|
3adant2 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 ∈ 𝐴 → ( 𝐶 +o 𝐵 ) ∈ ( 𝐶 +o 𝐴 ) ) ) |
7 |
4 6
|
orim12d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ) → ( ( 𝐶 +o 𝐴 ) = ( 𝐶 +o 𝐵 ) ∨ ( 𝐶 +o 𝐵 ) ∈ ( 𝐶 +o 𝐴 ) ) ) ) |
8 |
7
|
con3d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ¬ ( ( 𝐶 +o 𝐴 ) = ( 𝐶 +o 𝐵 ) ∨ ( 𝐶 +o 𝐵 ) ∈ ( 𝐶 +o 𝐴 ) ) → ¬ ( 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ) ) ) |
9 |
|
df-3an |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ↔ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ On ) ) |
10 |
|
ancom |
⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ On ) ↔ ( 𝐶 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ) |
11 |
|
anandi |
⊢ ( ( 𝐶 ∈ On ∧ ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ) ↔ ( ( 𝐶 ∈ On ∧ 𝐴 ∈ On ) ∧ ( 𝐶 ∈ On ∧ 𝐵 ∈ On ) ) ) |
12 |
9 10 11
|
3bitri |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ↔ ( ( 𝐶 ∈ On ∧ 𝐴 ∈ On ) ∧ ( 𝐶 ∈ On ∧ 𝐵 ∈ On ) ) ) |
13 |
|
oacl |
⊢ ( ( 𝐶 ∈ On ∧ 𝐴 ∈ On ) → ( 𝐶 +o 𝐴 ) ∈ On ) |
14 |
|
eloni |
⊢ ( ( 𝐶 +o 𝐴 ) ∈ On → Ord ( 𝐶 +o 𝐴 ) ) |
15 |
13 14
|
syl |
⊢ ( ( 𝐶 ∈ On ∧ 𝐴 ∈ On ) → Ord ( 𝐶 +o 𝐴 ) ) |
16 |
|
oacl |
⊢ ( ( 𝐶 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐶 +o 𝐵 ) ∈ On ) |
17 |
|
eloni |
⊢ ( ( 𝐶 +o 𝐵 ) ∈ On → Ord ( 𝐶 +o 𝐵 ) ) |
18 |
16 17
|
syl |
⊢ ( ( 𝐶 ∈ On ∧ 𝐵 ∈ On ) → Ord ( 𝐶 +o 𝐵 ) ) |
19 |
15 18
|
anim12i |
⊢ ( ( ( 𝐶 ∈ On ∧ 𝐴 ∈ On ) ∧ ( 𝐶 ∈ On ∧ 𝐵 ∈ On ) ) → ( Ord ( 𝐶 +o 𝐴 ) ∧ Ord ( 𝐶 +o 𝐵 ) ) ) |
20 |
12 19
|
sylbi |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( Ord ( 𝐶 +o 𝐴 ) ∧ Ord ( 𝐶 +o 𝐵 ) ) ) |
21 |
|
ordtri2 |
⊢ ( ( Ord ( 𝐶 +o 𝐴 ) ∧ Ord ( 𝐶 +o 𝐵 ) ) → ( ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ↔ ¬ ( ( 𝐶 +o 𝐴 ) = ( 𝐶 +o 𝐵 ) ∨ ( 𝐶 +o 𝐵 ) ∈ ( 𝐶 +o 𝐴 ) ) ) ) |
22 |
20 21
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ↔ ¬ ( ( 𝐶 +o 𝐴 ) = ( 𝐶 +o 𝐵 ) ∨ ( 𝐶 +o 𝐵 ) ∈ ( 𝐶 +o 𝐴 ) ) ) ) |
23 |
|
eloni |
⊢ ( 𝐴 ∈ On → Ord 𝐴 ) |
24 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
25 |
23 24
|
anim12i |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( Ord 𝐴 ∧ Ord 𝐵 ) ) |
26 |
25
|
3adant3 |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( Ord 𝐴 ∧ Ord 𝐵 ) ) |
27 |
|
ordtri2 |
⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ∈ 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ) ) ) |
28 |
26 27
|
syl |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴 ) ) ) |
29 |
8 22 28
|
3imtr4d |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) → 𝐴 ∈ 𝐵 ) ) |
30 |
2 29
|
impbid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 ↔ ( 𝐶 +o 𝐴 ) ∈ ( 𝐶 +o 𝐵 ) ) ) |