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Theorem oaass 7686
Description: Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. (Contributed by NM, 10-Dec-2004.)
Assertion
Ref Expression
oaass ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))

Proof of Theorem oaass
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . . 5 (𝑥 = ∅ → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 ∅))
2 oveq2 6698 . . . . . 6 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
32oveq2d 6706 . . . . 5 (𝑥 = ∅ → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
41, 3eqeq12d 2666 . . . 4 (𝑥 = ∅ → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅))))
5 oveq2 6698 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
6 oveq2 6698 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6706 . . . . 5 (𝑥 = 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
85, 7eqeq12d 2666 . . . 4 (𝑥 = 𝑦 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
9 oveq2 6698 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦))
10 oveq2 6698 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1110oveq2d 6706 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))
129, 11eqeq12d 2666 . . . 4 (𝑥 = suc 𝑦 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦))))
13 oveq2 6698 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝐶))
14 oveq2 6698 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1514oveq2d 6706 . . . . 5 (𝑥 = 𝐶 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))
1613, 15eqeq12d 2666 . . . 4 (𝑥 = 𝐶 → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
17 oacl 7660 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
18 oa0 7641 . . . . . 6 ((𝐴 +𝑜 𝐵) ∈ On → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
1917, 18syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 𝐵))
20 oa0 7641 . . . . . . 7 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
2120oveq2d 6706 . . . . . 6 (𝐵 ∈ On → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
2221adantl 481 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 ∅)) = (𝐴 +𝑜 𝐵))
2319, 22eqtr4d 2688 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
24 suceq 5828 . . . . . 6 (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
25 oasuc 7649 . . . . . . . 8 (((𝐴 +𝑜 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
2617, 25sylan 487 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
27 oasuc 7649 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
2827oveq2d 6706 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
2928adantl 481 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)))
30 oacl 7660 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
31 oasuc 7649 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3230, 31sylan2 490 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 suc (𝐵 +𝑜 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3329, 32eqtrd 2685 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3433anassrs 681 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3526, 34eqeq12d 2666 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) ↔ suc ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
3624, 35syl5ibr 236 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦))))
3736expcom 450 . . . 4 (𝑦 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))))
38 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
39 oalim 7657 . . . . . . . . . 10 (((𝐴 +𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
4038, 39mpanr1 719 . . . . . . . . 9 (((𝐴 +𝑜 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
4117, 40sylan 487 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
4241ancoms 468 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
4342adantr 480 . . . . . 6 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
44 oalimcl 7685 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +𝑜 𝑥))
4538, 44mpanr1 719 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +𝑜 𝑥))
4645ancoms 468 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +𝑜 𝑥))
47 ovex 6718 . . . . . . . . . . . 12 (𝐵 +𝑜 𝑥) ∈ V
48 oalim 7657 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ ((𝐵 +𝑜 𝑥) ∈ V ∧ Lim (𝐵 +𝑜 𝑥))) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
4947, 48mpanr1 719 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ Lim (𝐵 +𝑜 𝑥)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
5046, 49sylan2 490 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
51 limelon 5826 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5238, 51mpan 706 . . . . . . . . . . . . . . . . 17 (Lim 𝑥𝑥 ∈ On)
53 oacl 7660 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
5453ancoms 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
55 onelon 5786 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵 +𝑜 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On)
5655ex 449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 +𝑜 𝑥) ∈ On → (𝑧 ∈ (𝐵 +𝑜 𝑥) → 𝑧 ∈ On))
5754, 56syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → 𝑧 ∈ On))
5857adantld 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On))
5958adantl 481 . . . . . . . . . . . . . . . . . . 19 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ∈ On))
60 0ellim 5825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑥 → ∅ ∈ 𝑥)
61 onelss 5804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐵 ∈ On → (𝑧𝐵𝑧𝐵))
6220sseq2d 3666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +𝑜 ∅) ↔ 𝑧𝐵))
6361, 62sylibrd 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧𝐵𝑧 ⊆ (𝐵 +𝑜 ∅)))
6463imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐵 ∈ On ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐵 +𝑜 ∅))
65 oveq2 6698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → (𝐵 +𝑜 𝑦) = (𝐵 +𝑜 ∅))
6665sseq2d 3666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ 𝑧 ⊆ (𝐵 +𝑜 ∅)))
6766rspcev 3340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((∅ ∈ 𝑥𝑧 ⊆ (𝐵 +𝑜 ∅)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
6860, 64, 67syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧𝐵)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
6968expr 642 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥𝐵 ∈ On) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
7069adantrl 752 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
7170adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
72 oawordex 7682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧))
7372ad2ant2l 797 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧))
74 oaord 7672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 ↔ (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
75743expb 1285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥 ↔ (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
76 eleq1 2718 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐵 +𝑜 𝑦) = 𝑧 → ((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
7775, 76sylan9bb 736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑦) = 𝑧) → (𝑦𝑥𝑧 ∈ (𝐵 +𝑜 𝑥)))
7877an32s 863 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥𝑧 ∈ (𝐵 +𝑜 𝑥)))
7978biimpar 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑦𝑥)
80 eqimss2 3691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐵 +𝑜 𝑦) = 𝑧𝑧 ⊆ (𝐵 +𝑜 𝑦))
8180ad3antlr 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → 𝑧 ⊆ (𝐵 +𝑜 𝑦))
8279, 81jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑦𝑥𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8382anasss 680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥))) → (𝑦𝑥𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8483expcom 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +𝑜 𝑦) = 𝑧) → (𝑦𝑥𝑧 ⊆ (𝐵 +𝑜 𝑦))))
8584reximdv2 3043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8685adantrr 753 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8773, 86sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8887adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
89 eloni 5771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ On → Ord 𝑧)
90 eloni 5771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐵 ∈ On → Ord 𝐵)
91 ordtri2or 5860 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧𝐵𝐵𝑧))
9289, 90, 91syl2anr 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧𝐵𝐵𝑧))
9392ad2ant2l 797 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝑧𝐵𝐵𝑧))
9493adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵𝐵𝑧))
9571, 88, 94mpjaod 395 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
9695exp45 641 . . . . . . . . . . . . . . . . . . . . . . . 24 (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))))
9796imp 444 . . . . . . . . . . . . . . . . . . . . . . 23 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))))
9897adantld 482 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))))
9998imp32 448 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦))
100 simplrr 818 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝑧 ∈ On)
101 onelon 5786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
102101, 30sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵 +𝑜 𝑦) ∈ On)
103102exp32 630 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐵 ∈ On → (𝑥 ∈ On → (𝑦𝑥 → (𝐵 +𝑜 𝑦) ∈ On)))
104103com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ On → (𝐵 ∈ On → (𝑦𝑥 → (𝐵 +𝑜 𝑦) ∈ On)))
105104imp31 447 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
106105adantll 750 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
107106adantlr 751 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
108 simpll 805 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
109108ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
110 oaword 7674 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ On ∧ (𝐵 +𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
111100, 107, 109, 110syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
112111rexbidva 3078 . . . . . . . . . . . . . . . . . . . . 21 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) → (∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
11399, 112mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
114113exp32 630 . . . . . . . . . . . . . . . . . . 19 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (𝑧 ∈ On → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))))
11559, 114mpdd 43 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
116115exp32 630 . . . . . . . . . . . . . . . . 17 (Lim 𝑥 → (𝑥 ∈ On → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))))
11752, 116mpd 15 . . . . . . . . . . . . . . . 16 (Lim 𝑥 → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))))
118117exp4a 632 . . . . . . . . . . . . . . 15 (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +𝑜 𝑥) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))))
119118imp31 447 . . . . . . . . . . . . . 14 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +𝑜 𝑥) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
120119ralrimiv 2994 . . . . . . . . . . . . 13 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +𝑜 𝑥)∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
121 iunss2 4597 . . . . . . . . . . . . 13 (∀𝑧 ∈ (𝐵 +𝑜 𝑥)∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ⊆ 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
122120, 121syl 17 . . . . . . . . . . . 12 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ⊆ 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
123122ancoms 468 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ⊆ 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
124 oaordi 7671 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
125124anim1d 587 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦𝑥𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))))
126 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
127126eleq2d 2716 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +𝑜 𝑦) → (𝑤 ∈ (𝐴 +𝑜 𝑧) ↔ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
128127rspcev 3340 . . . . . . . . . . . . . . . . . 18 (((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))
129125, 128syl6 35 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦𝑥𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧)))
130129expd 451 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))))
131130rexlimdv 3059 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦𝑥 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧)))
132 eliun 4556 . . . . . . . . . . . . . . 15 (𝑤 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ↔ ∃𝑦𝑥 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
133 eliun 4556 . . . . . . . . . . . . . . 15 (𝑤 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ↔ ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))
134131, 132, 1333imtr4g 285 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → 𝑤 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧)))
135134ssrdv 3642 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ⊆ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
13652, 135sylan 487 . . . . . . . . . . . 12 ((Lim 𝑥𝐵 ∈ On) → 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ⊆ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
137136adantl 481 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ⊆ 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧))
138123, 137eqssd 3653 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
13950, 138eqtrd 2685 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
140139an12s 860 . . . . . . . 8 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
141140adantr 480 . . . . . . 7 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
142 iuneq2 4569 . . . . . . . 8 (∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
143142adantl 481 . . . . . . 7 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
144141, 143eqtr4d 2688 . . . . . 6 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
14543, 144eqtr4d 2688 . . . . 5 (((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ ∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)))
146145exp31 629 . . . 4 (Lim 𝑥 → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)))))
1474, 8, 12, 16, 23, 37, 146tfinds3 7106 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
148147com12 32 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶))))
1491483impia 1280 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 𝐶) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948   ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690   +𝑜 coa 7602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609
This theorem is referenced by:  odi  7704  oaabs  7769  oaabs2  7770
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