Step | Hyp | Ref
| Expression |
1 | | oveq1 7365 |
. . . 4
⊢ (𝑎 = 𝑥 → (𝑎 +no 𝑏) = (𝑥 +no 𝑏)) |
2 | 1 | oveq1d 7373 |
. . 3
⊢ (𝑎 = 𝑥 → ((𝑎 +no 𝑏) +no 𝑐) = ((𝑥 +no 𝑏) +no 𝑐)) |
3 | | oveq1 7365 |
. . 3
⊢ (𝑎 = 𝑥 → (𝑎 +no (𝑏 +no 𝑐)) = (𝑥 +no (𝑏 +no 𝑐))) |
4 | 2, 3 | eqeq12d 2753 |
. 2
⊢ (𝑎 = 𝑥 → (((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐)) ↔ ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)))) |
5 | | oveq2 7366 |
. . . 4
⊢ (𝑏 = 𝑦 → (𝑥 +no 𝑏) = (𝑥 +no 𝑦)) |
6 | 5 | oveq1d 7373 |
. . 3
⊢ (𝑏 = 𝑦 → ((𝑥 +no 𝑏) +no 𝑐) = ((𝑥 +no 𝑦) +no 𝑐)) |
7 | | oveq1 7365 |
. . . 4
⊢ (𝑏 = 𝑦 → (𝑏 +no 𝑐) = (𝑦 +no 𝑐)) |
8 | 7 | oveq2d 7374 |
. . 3
⊢ (𝑏 = 𝑦 → (𝑥 +no (𝑏 +no 𝑐)) = (𝑥 +no (𝑦 +no 𝑐))) |
9 | 6, 8 | eqeq12d 2753 |
. 2
⊢ (𝑏 = 𝑦 → (((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ↔ ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)))) |
10 | | oveq2 7366 |
. . 3
⊢ (𝑐 = 𝑧 → ((𝑥 +no 𝑦) +no 𝑐) = ((𝑥 +no 𝑦) +no 𝑧)) |
11 | | oveq2 7366 |
. . . 4
⊢ (𝑐 = 𝑧 → (𝑦 +no 𝑐) = (𝑦 +no 𝑧)) |
12 | 11 | oveq2d 7374 |
. . 3
⊢ (𝑐 = 𝑧 → (𝑥 +no (𝑦 +no 𝑐)) = (𝑥 +no (𝑦 +no 𝑧))) |
13 | 10, 12 | eqeq12d 2753 |
. 2
⊢ (𝑐 = 𝑧 → (((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ↔ ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)))) |
14 | | oveq1 7365 |
. . . 4
⊢ (𝑎 = 𝑥 → (𝑎 +no 𝑦) = (𝑥 +no 𝑦)) |
15 | 14 | oveq1d 7373 |
. . 3
⊢ (𝑎 = 𝑥 → ((𝑎 +no 𝑦) +no 𝑧) = ((𝑥 +no 𝑦) +no 𝑧)) |
16 | | oveq1 7365 |
. . 3
⊢ (𝑎 = 𝑥 → (𝑎 +no (𝑦 +no 𝑧)) = (𝑥 +no (𝑦 +no 𝑧))) |
17 | 15, 16 | eqeq12d 2753 |
. 2
⊢ (𝑎 = 𝑥 → (((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ↔ ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)))) |
18 | | oveq2 7366 |
. . . 4
⊢ (𝑏 = 𝑦 → (𝑎 +no 𝑏) = (𝑎 +no 𝑦)) |
19 | 18 | oveq1d 7373 |
. . 3
⊢ (𝑏 = 𝑦 → ((𝑎 +no 𝑏) +no 𝑧) = ((𝑎 +no 𝑦) +no 𝑧)) |
20 | | oveq1 7365 |
. . . 4
⊢ (𝑏 = 𝑦 → (𝑏 +no 𝑧) = (𝑦 +no 𝑧)) |
21 | 20 | oveq2d 7374 |
. . 3
⊢ (𝑏 = 𝑦 → (𝑎 +no (𝑏 +no 𝑧)) = (𝑎 +no (𝑦 +no 𝑧))) |
22 | 19, 21 | eqeq12d 2753 |
. 2
⊢ (𝑏 = 𝑦 → (((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)) ↔ ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)))) |
23 | 5 | oveq1d 7373 |
. . 3
⊢ (𝑏 = 𝑦 → ((𝑥 +no 𝑏) +no 𝑧) = ((𝑥 +no 𝑦) +no 𝑧)) |
24 | 20 | oveq2d 7374 |
. . 3
⊢ (𝑏 = 𝑦 → (𝑥 +no (𝑏 +no 𝑧)) = (𝑥 +no (𝑦 +no 𝑧))) |
25 | 23, 24 | eqeq12d 2753 |
. 2
⊢ (𝑏 = 𝑦 → (((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧)) ↔ ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)))) |
26 | | oveq2 7366 |
. . 3
⊢ (𝑐 = 𝑧 → ((𝑎 +no 𝑦) +no 𝑐) = ((𝑎 +no 𝑦) +no 𝑧)) |
27 | 11 | oveq2d 7374 |
. . 3
⊢ (𝑐 = 𝑧 → (𝑎 +no (𝑦 +no 𝑐)) = (𝑎 +no (𝑦 +no 𝑧))) |
28 | 26, 27 | eqeq12d 2753 |
. 2
⊢ (𝑐 = 𝑧 → (((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)) ↔ ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)))) |
29 | | oveq1 7365 |
. . . 4
⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏)) |
30 | 29 | oveq1d 7373 |
. . 3
⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) +no 𝑐) = ((𝐴 +no 𝑏) +no 𝑐)) |
31 | | oveq1 7365 |
. . 3
⊢ (𝑎 = 𝐴 → (𝑎 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝑏 +no 𝑐))) |
32 | 30, 31 | eqeq12d 2753 |
. 2
⊢ (𝑎 = 𝐴 → (((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐)) ↔ ((𝐴 +no 𝑏) +no 𝑐) = (𝐴 +no (𝑏 +no 𝑐)))) |
33 | | oveq2 7366 |
. . . 4
⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵)) |
34 | 33 | oveq1d 7373 |
. . 3
⊢ (𝑏 = 𝐵 → ((𝐴 +no 𝑏) +no 𝑐) = ((𝐴 +no 𝐵) +no 𝑐)) |
35 | | oveq1 7365 |
. . . 4
⊢ (𝑏 = 𝐵 → (𝑏 +no 𝑐) = (𝐵 +no 𝑐)) |
36 | 35 | oveq2d 7374 |
. . 3
⊢ (𝑏 = 𝐵 → (𝐴 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝐵 +no 𝑐))) |
37 | 34, 36 | eqeq12d 2753 |
. 2
⊢ (𝑏 = 𝐵 → (((𝐴 +no 𝑏) +no 𝑐) = (𝐴 +no (𝑏 +no 𝑐)) ↔ ((𝐴 +no 𝐵) +no 𝑐) = (𝐴 +no (𝐵 +no 𝑐)))) |
38 | | oveq2 7366 |
. . 3
⊢ (𝑐 = 𝐶 → ((𝐴 +no 𝐵) +no 𝑐) = ((𝐴 +no 𝐵) +no 𝐶)) |
39 | | oveq2 7366 |
. . . 4
⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶)) |
40 | 39 | oveq2d 7374 |
. . 3
⊢ (𝑐 = 𝐶 → (𝐴 +no (𝐵 +no 𝑐)) = (𝐴 +no (𝐵 +no 𝐶))) |
41 | 38, 40 | eqeq12d 2753 |
. 2
⊢ (𝑐 = 𝐶 → (((𝐴 +no 𝐵) +no 𝑐) = (𝐴 +no (𝐵 +no 𝑐)) ↔ ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶)))) |
42 | | simpr21 1261 |
. . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐))) |
43 | | eleq1 2826 |
. . . . . . . . 9
⊢ (((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) → (((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤)) |
44 | 43 | ralimi 3087 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) → ∀𝑥 ∈ 𝑎 (((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤)) |
45 | | ralbi 3107 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝑎 (((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤) → (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ ∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤)) |
46 | 42, 44, 45 | 3syl 18 |
. . . . . . 7
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ↔ ∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤)) |
47 | | simpr23 1263 |
. . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) |
48 | | eleq1 2826 |
. . . . . . . . 9
⊢ (((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)) → (((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤)) |
49 | 48 | ralimi 3087 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐)) → ∀𝑦 ∈ 𝑏 (((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤)) |
50 | | ralbi 3107 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝑏 (((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤) → (∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤)) |
51 | 47, 49, 50 | 3syl 18 |
. . . . . . 7
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ↔ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤)) |
52 | | simpr3 1197 |
. . . . . . . 8
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧))) |
53 | | eleq1 2826 |
. . . . . . . . 9
⊢ (((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)) → (((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)) |
54 | 53 | ralimi 3087 |
. . . . . . . 8
⊢
(∀𝑧 ∈
𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)) → ∀𝑧 ∈ 𝑐 (((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)) |
55 | | ralbi 3107 |
. . . . . . . 8
⊢
(∀𝑧 ∈
𝑐 (((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤) → (∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)) |
56 | 52, 54, 55 | 3syl 18 |
. . . . . . 7
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤 ↔ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)) |
57 | 46, 51, 56 | 3anbi123d 1437 |
. . . . . 6
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ((∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤) ↔ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤))) |
58 | 57 | rabbidv 3416 |
. . . . 5
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)} = {𝑤 ∈ On ∣ (∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)}) |
59 | 58 | inteqd 4913 |
. . . 4
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ∩
{𝑤 ∈ On ∣
(∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)} = ∩ {𝑤 ∈ On ∣
(∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)}) |
60 | | naddasslem1 8639 |
. . . . 5
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → ((𝑎 +no 𝑏) +no 𝑐) = ∩ {𝑤 ∈ On ∣
(∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)}) |
61 | 60 | adantr 482 |
. . . 4
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ((𝑎 +no 𝑏) +no 𝑐) = ∩ {𝑤 ∈ On ∣
(∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) ∈ 𝑤)}) |
62 | | naddasslem2 8640 |
. . . . 5
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (𝑎 +no (𝑏 +no 𝑐)) = ∩ {𝑤 ∈ On ∣
(∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)}) |
63 | 62 | adantr 482 |
. . . 4
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → (𝑎 +no (𝑏 +no 𝑐)) = ∩ {𝑤 ∈ On ∣
(∀𝑥 ∈ 𝑎 (𝑥 +no (𝑏 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑦 ∈ 𝑏 (𝑎 +no (𝑦 +no 𝑐)) ∈ 𝑤 ∧ ∀𝑧 ∈ 𝑐 (𝑎 +no (𝑏 +no 𝑧)) ∈ 𝑤)}) |
64 | 59, 61, 63 | 3eqtr4d 2787 |
. . 3
⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) ∧
((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧)))) → ((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐))) |
65 | 64 | ex 414 |
. 2
⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) →
(((∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑦) +no 𝑧) = (𝑥 +no (𝑦 +no 𝑧)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑥 +no 𝑦) +no 𝑐) = (𝑥 +no (𝑦 +no 𝑐)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑧 ∈ 𝑐 ((𝑥 +no 𝑏) +no 𝑧) = (𝑥 +no (𝑏 +no 𝑧))) ∧ (∀𝑥 ∈ 𝑎 ((𝑥 +no 𝑏) +no 𝑐) = (𝑥 +no (𝑏 +no 𝑐)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑦) +no 𝑧) = (𝑎 +no (𝑦 +no 𝑧)) ∧ ∀𝑦 ∈ 𝑏 ((𝑎 +no 𝑦) +no 𝑐) = (𝑎 +no (𝑦 +no 𝑐))) ∧ ∀𝑧 ∈ 𝑐 ((𝑎 +no 𝑏) +no 𝑧) = (𝑎 +no (𝑏 +no 𝑧))) → ((𝑎 +no 𝑏) +no 𝑐) = (𝑎 +no (𝑏 +no 𝑐)))) |
66 | 4, 9, 13, 17, 22, 25, 28, 32, 37, 41, 65 | on3ind 8617 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = (𝐴 +no (𝐵 +no 𝐶))) |