Proof of Theorem oecan
Step | Hyp | Ref
| Expression |
1 | | oeordi 8237 |
. . . . . . 7
⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ (On ∖
2o)) → (𝐵
∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶))) |
2 | 1 | ancoms 462 |
. . . . . 6
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐶
∈ On) → (𝐵 ∈
𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶))) |
3 | 2 | 3adant2 1132 |
. . . . 5
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → (𝐵 ∈ 𝐶 → (𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶))) |
4 | | oeordi 8237 |
. . . . . . 7
⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (On ∖
2o)) → (𝐶
∈ 𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))) |
5 | 4 | ancoms 462 |
. . . . . 6
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On) → (𝐶 ∈
𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))) |
6 | 5 | 3adant3 1133 |
. . . . 5
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → (𝐶 ∈ 𝐵 → (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵))) |
7 | 3, 6 | orim12d 964 |
. . . 4
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → ((𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵) → ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))) |
8 | 7 | con3d 155 |
. . 3
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → (¬ ((𝐴
↑o 𝐵)
∈ (𝐴
↑o 𝐶) ∨
(𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)) → ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
9 | | eldifi 4015 |
. . . . . 6
⊢ (𝐴 ∈ (On ∖
2o) → 𝐴
∈ On) |
10 | 9 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → 𝐴 ∈
On) |
11 | | simp2 1138 |
. . . . 5
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → 𝐵 ∈
On) |
12 | | oecl 8186 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) |
13 | 10, 11, 12 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → (𝐴
↑o 𝐵)
∈ On) |
14 | | simp3 1139 |
. . . . 5
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → 𝐶 ∈
On) |
15 | | oecl 8186 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o 𝐶) ∈ On) |
16 | 10, 14, 15 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → (𝐴
↑o 𝐶)
∈ On) |
17 | | eloni 6176 |
. . . . 5
⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) |
18 | | eloni 6176 |
. . . . 5
⊢ ((𝐴 ↑o 𝐶) ∈ On → Ord (𝐴 ↑o 𝐶)) |
19 | | ordtri3 6202 |
. . . . 5
⊢ ((Ord
(𝐴 ↑o 𝐵) ∧ Ord (𝐴 ↑o 𝐶)) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ ¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))) |
20 | 17, 18, 19 | syl2an 599 |
. . . 4
⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ (𝐴 ↑o 𝐶) ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ ¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))) |
21 | 13, 16, 20 | syl2anc 587 |
. . 3
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → ((𝐴
↑o 𝐵) =
(𝐴 ↑o 𝐶) ↔ ¬ ((𝐴 ↑o 𝐵) ∈ (𝐴 ↑o 𝐶) ∨ (𝐴 ↑o 𝐶) ∈ (𝐴 ↑o 𝐵)))) |
22 | | eloni 6176 |
. . . . 5
⊢ (𝐵 ∈ On → Ord 𝐵) |
23 | | eloni 6176 |
. . . . 5
⊢ (𝐶 ∈ On → Ord 𝐶) |
24 | | ordtri3 6202 |
. . . . 5
⊢ ((Ord
𝐵 ∧ Ord 𝐶) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
25 | 22, 23, 24 | syl2an 599 |
. . . 4
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
26 | 25 | 3adant1 1131 |
. . 3
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → (𝐵 = 𝐶 ↔ ¬ (𝐵 ∈ 𝐶 ∨ 𝐶 ∈ 𝐵))) |
27 | 8, 21, 26 | 3imtr4d 297 |
. 2
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → ((𝐴
↑o 𝐵) =
(𝐴 ↑o 𝐶) → 𝐵 = 𝐶)) |
28 | | oveq2 7172 |
. 2
⊢ (𝐵 = 𝐶 → (𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶)) |
29 | 27, 28 | impbid1 228 |
1
⊢ ((𝐴 ∈ (On ∖
2o) ∧ 𝐵
∈ On ∧ 𝐶 ∈
On) → ((𝐴
↑o 𝐵) =
(𝐴 ↑o 𝐶) ↔ 𝐵 = 𝐶)) |