| Step | Hyp | Ref
| Expression |
| 1 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = ∅ → (𝐴 +no 𝑦) = (𝐴 +no ∅)) |
| 2 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = ∅ → (𝐴 +o 𝑦) = (𝐴 +o ∅)) |
| 3 | 1, 2 | eqeq12d 2753 |
. . . 4
⊢ (𝑦 = ∅ → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no ∅) = (𝐴 +o ∅))) |
| 4 | 3 | imbi2d 340 |
. . 3
⊢ (𝑦 = ∅ → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o ∅)))) |
| 5 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no 𝑥)) |
| 6 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o 𝑥)) |
| 7 | 5, 6 | eqeq12d 2753 |
. . . 4
⊢ (𝑦 = 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝑥) = (𝐴 +o 𝑥))) |
| 8 | 7 | imbi2d 340 |
. . 3
⊢ (𝑦 = 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥)))) |
| 9 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = suc 𝑥 → (𝐴 +no 𝑦) = (𝐴 +no suc 𝑥)) |
| 10 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = suc 𝑥 → (𝐴 +o 𝑦) = (𝐴 +o suc 𝑥)) |
| 11 | 9, 10 | eqeq12d 2753 |
. . . 4
⊢ (𝑦 = suc 𝑥 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥))) |
| 12 | 11 | imbi2d 340 |
. . 3
⊢ (𝑦 = suc 𝑥 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)))) |
| 13 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = 𝐵 → (𝐴 +no 𝑦) = (𝐴 +no 𝐵)) |
| 14 | | oveq2 7439 |
. . . . 5
⊢ (𝑦 = 𝐵 → (𝐴 +o 𝑦) = (𝐴 +o 𝐵)) |
| 15 | 13, 14 | eqeq12d 2753 |
. . . 4
⊢ (𝑦 = 𝐵 → ((𝐴 +no 𝑦) = (𝐴 +o 𝑦) ↔ (𝐴 +no 𝐵) = (𝐴 +o 𝐵))) |
| 16 | 15 | imbi2d 340 |
. . 3
⊢ (𝑦 = 𝐵 → ((𝐴 ∈ On → (𝐴 +no 𝑦) = (𝐴 +o 𝑦)) ↔ (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵)))) |
| 17 | | naddrid 8721 |
. . . 4
⊢ (𝐴 ∈ On → (𝐴 +no ∅) = 𝐴) |
| 18 | | oa0 8554 |
. . . 4
⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
| 19 | 17, 18 | eqtr4d 2780 |
. . 3
⊢ (𝐴 ∈ On → (𝐴 +no ∅) = (𝐴 +o
∅)) |
| 20 | | suceq 6450 |
. . . . . . 7
⊢ ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥)) |
| 21 | 20 | 3ad2ant3 1136 |
. . . . . 6
⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → suc (𝐴 +no 𝑥) = suc (𝐴 +o 𝑥)) |
| 22 | | nnon 7893 |
. . . . . . . . 9
⊢ (𝑥 ∈ ω → 𝑥 ∈ On) |
| 23 | | naddsuc2 8739 |
. . . . . . . . 9
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥)) |
| 24 | 22, 23 | sylan2 593 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥)) |
| 25 | 24 | ancoms 458 |
. . . . . . 7
⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥)) |
| 26 | 25 | 3adant3 1133 |
. . . . . 6
⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = suc (𝐴 +no 𝑥)) |
| 27 | | onasuc 8566 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥)) |
| 28 | 27 | ancoms 458 |
. . . . . . 7
⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥)) |
| 29 | 28 | 3adant3 1133 |
. . . . . 6
⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥)) |
| 30 | 21, 26, 29 | 3eqtr4d 2787 |
. . . . 5
⊢ ((𝑥 ∈ ω ∧ 𝐴 ∈ On ∧ (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)) |
| 31 | 30 | 3exp 1120 |
. . . 4
⊢ (𝑥 ∈ ω → (𝐴 ∈ On → ((𝐴 +no 𝑥) = (𝐴 +o 𝑥) → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)))) |
| 32 | 31 | a2d 29 |
. . 3
⊢ (𝑥 ∈ ω → ((𝐴 ∈ On → (𝐴 +no 𝑥) = (𝐴 +o 𝑥)) → (𝐴 ∈ On → (𝐴 +no suc 𝑥) = (𝐴 +o suc 𝑥)))) |
| 33 | 4, 8, 12, 16, 19, 32 | finds 7918 |
. 2
⊢ (𝐵 ∈ ω → (𝐴 ∈ On → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))) |
| 34 | 33 | impcom 407 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵)) |