Step | Hyp | Ref
| Expression |
1 | | oveq1 7239 |
. . 3
⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏)) |
2 | | oveq2 7240 |
. . 3
⊢ (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐)) |
3 | 1, 2 | eqeq12d 2754 |
. 2
⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐))) |
4 | | oveq2 7240 |
. . 3
⊢ (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑)) |
5 | | oveq1 7239 |
. . 3
⊢ (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐)) |
6 | 4, 5 | eqeq12d 2754 |
. 2
⊢ (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐))) |
7 | | oveq1 7239 |
. . 3
⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑)) |
8 | | oveq2 7240 |
. . 3
⊢ (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐)) |
9 | 7, 8 | eqeq12d 2754 |
. 2
⊢ (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐))) |
10 | | oveq1 7239 |
. . 3
⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏)) |
11 | | oveq2 7240 |
. . 3
⊢ (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴)) |
12 | 10, 11 | eqeq12d 2754 |
. 2
⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴))) |
13 | | oveq2 7240 |
. . 3
⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵)) |
14 | | oveq1 7239 |
. . 3
⊢ (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴)) |
15 | 13, 14 | eqeq12d 2754 |
. 2
⊢ (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴))) |
16 | | eleq1 2826 |
. . . . . . . . . . . 12
⊢ ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥)) |
17 | 16 | ralimi 3085 |
. . . . . . . . . . 11
⊢
(∀𝑑 ∈
𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑 ∈ 𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥)) |
18 | | ralbi 3091 |
. . . . . . . . . . 11
⊢
(∀𝑑 ∈
𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)) |
19 | 17, 18 | syl 17 |
. . . . . . . . . 10
⊢
(∀𝑑 ∈
𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)) |
20 | 19 | 3ad2ant3 1137 |
. . . . . . . . 9
⊢
((∀𝑐 ∈
𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)) |
21 | 20 | adantl 485 |
. . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)) |
22 | | eleq1 2826 |
. . . . . . . . . . . 12
⊢ ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥)) |
23 | 22 | ralimi 3085 |
. . . . . . . . . . 11
⊢
(∀𝑐 ∈
𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐 ∈ 𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥)) |
24 | | ralbi 3091 |
. . . . . . . . . . 11
⊢
(∀𝑐 ∈
𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥)) |
25 | 23, 24 | syl 17 |
. . . . . . . . . 10
⊢
(∀𝑐 ∈
𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥)) |
26 | 25 | 3ad2ant2 1136 |
. . . . . . . . 9
⊢
((∀𝑐 ∈
𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥)) |
27 | 26 | adantl 485 |
. . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥)) |
28 | 21, 27 | anbi12d 634 |
. . . . . . 7
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥))) |
29 | 28 | biancomd 467 |
. . . . . 6
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥))) |
30 | 29 | rabbidv 3403 |
. . . . 5
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → {𝑥 ∈ On ∣ (∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)}) |
31 | 30 | inteqd 4879 |
. . . 4
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ∩
{𝑥 ∈ On ∣
(∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = ∩ {𝑥 ∈ On ∣
(∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)}) |
32 | | naddov2 33600 |
. . . . 5
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +no 𝑏) = ∩ {𝑥 ∈ On ∣
(∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)}) |
33 | 32 | adantr 484 |
. . . 4
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = ∩ {𝑥 ∈ On ∣
(∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) ∈ 𝑥)}) |
34 | | naddov2 33600 |
. . . . . 6
⊢ ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 +no 𝑎) = ∩ {𝑥 ∈ On ∣
(∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)}) |
35 | 34 | ancoms 462 |
. . . . 5
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +no 𝑎) = ∩ {𝑥 ∈ On ∣
(∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)}) |
36 | 35 | adantr 484 |
. . . 4
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑏 +no 𝑎) = ∩ {𝑥 ∈ On ∣
(∀𝑐 ∈ 𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑 ∈ 𝑏 (𝑑 +no 𝑎) ∈ 𝑥)}) |
37 | 31, 33, 36 | 3eqtr4d 2788 |
. . 3
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎)) |
38 | 37 | ex 416 |
. 2
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) →
((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎))) |
39 | 3, 6, 9, 12, 15, 38 | on2ind 33594 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴)) |