Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oaass Structured version   Visualization version   GIF version

Theorem oaass 7528
 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 6557 . . . . 5 (𝑥 = ∅ → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 ∅))
2 oveq2 6557 . . . . . 6 (𝑥 = ∅ → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 ∅))
32oveq2d 6565 . . . . 5 (𝑥 = ∅ → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
41, 3eqeq12d 2625 . . . 4 (𝑥 = ∅ → (((𝐴 +𝑜 𝐵) +𝑜 𝑥) = (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) ↔ ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅))))
5 oveq2 6557 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
6 oveq2 6557 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝑦))
76oveq2d 6565 . . . . 5 (𝑥 = 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
85, 7eqeq12d 2625 . . . 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 6557 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 suc 𝑦))
10 oveq2 6557 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 suc 𝑦))
1110oveq2d 6565 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)))
129, 11eqeq12d 2625 . . . 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 6557 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = ((𝐴 +𝑜 𝐵) +𝑜 𝐶))
14 oveq2 6557 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +𝑜 𝑥) = (𝐵 +𝑜 𝐶))
1514oveq2d 6565 . . . . 5 (𝑥 = 𝐶 → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = (𝐴 +𝑜 (𝐵 +𝑜 𝐶)))
1613, 15eqeq12d 2625 . . . 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 7502 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) ∈ On)
18 oa0 7483 . . . . . 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 7483 . . . . . . 7 (𝐵 ∈ On → (𝐵 +𝑜 ∅) = 𝐵)
2120oveq2d 6565 . . . . . 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 2647 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +𝑜 𝐵) +𝑜 ∅) = (𝐴 +𝑜 (𝐵 +𝑜 ∅)))
24 suceq 5707 . . . . . 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 7491 . . . . . . . 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 7491 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 suc 𝑦) = suc (𝐵 +𝑜 𝑦))
2827oveq2d 6565 . . . . . . . . . 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 7502 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +𝑜 𝑦) ∈ On)
31 oasuc 7491 . . . . . . . . . 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 2644 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3433anassrs 678 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +𝑜 (𝐵 +𝑜 suc 𝑦)) = suc (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
3526, 34eqeq12d 2625 . . . . . 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 235 . . . . 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 3176 . . . . . . . . . 10 𝑥 ∈ V
39 oalim 7499 . . . . . . . . . 10 (((𝐴 +𝑜 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +𝑜 𝐵) +𝑜 𝑥) = 𝑦𝑥 ((𝐴 +𝑜 𝐵) +𝑜 𝑦))
4038, 39mpanr1 715 . . . . . . . . 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 7527 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +𝑜 𝑥))
4538, 44mpanr1 715 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +𝑜 𝑥))
4645ancoms 468 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +𝑜 𝑥))
47 ovex 6577 . . . . . . . . . . . 12 (𝐵 +𝑜 𝑥) ∈ V
48 oalim 7499 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 5705 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5238, 51mpan 702 . . . . . . . . . . . . . . . . 17 (Lim 𝑥𝑥 ∈ On)
53 oacl 7502 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
5453ancoms 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝑥) ∈ On)
55 onelon 5665 . . . . . . . . . . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑥 → ∅ ∈ 𝑥)
61 onelss 5683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐵 ∈ On → (𝑧𝐵𝑧𝐵))
6220sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +𝑜 ∅) ↔ 𝑧𝐵))
6361, 62sylibrd 248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧𝐵𝑧 ⊆ (𝐵 +𝑜 ∅)))
6463imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝐵 ∈ On ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐵 +𝑜 ∅))
65 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → (𝐵 +𝑜 𝑦) = (𝐵 +𝑜 ∅))
6665sseq2d 3596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +𝑜 𝑦) ↔ 𝑧 ⊆ (𝐵 +𝑜 ∅)))
6766rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 641 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥𝐵 ∈ On) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
7069adantrl 748 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
7170adantrr 749 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
72 oawordex 7524 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧))
7372ad2ant2l 778 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧))
74 oaord 7514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 ↔ (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
75743expb 1258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥 ↔ (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
76 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐵 +𝑜 𝑦) = 𝑧 → ((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ↔ 𝑧 ∈ (𝐵 +𝑜 𝑥)))
7775, 76sylan9bb 732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +𝑜 𝑦) = 𝑧) → (𝑦𝑥𝑧 ∈ (𝐵 +𝑜 𝑥)))
7877an32s 842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝐵 +𝑜 𝑦) = 𝑧𝑧 ⊆ (𝐵 +𝑜 𝑦))
8180ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → (∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8685adantrr 749 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +𝑜 𝑥) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ On (𝐵 +𝑜 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +𝑜 𝑦)))
8773, 86sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑧 ∈ On → Ord 𝑧)
90 eloni 5650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐵 ∈ On → Ord 𝐵)
91 ordtri2or 5739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧𝐵𝐵𝑧))
9289, 90, 91syl2anr 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧𝐵𝐵𝑧))
9392ad2ant2l 778 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 640 . . . . . . . . . . . . . . . . . . . . . . . 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 797 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝑧 ∈ On)
101 onelon 5665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
102101, 30sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵 +𝑜 𝑦) ∈ On)
103102exp32 629 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
107106adantlr 747 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → (𝐵 +𝑜 𝑦) ∈ On)
108 simpll 786 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
109108ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
110 oaword 7516 . . . . . . . . . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . . . . . . . . . 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 3031 . . . . . . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . . . . . . 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 629 . . . . . . . . . . . . . . . . . . 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 42 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +𝑜 𝑥)) → ∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
116115exp32 629 . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . 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 2948 . . . . . . . . . . . . 13 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +𝑜 𝑥)∃𝑦𝑥 (𝐴 +𝑜 𝑧) ⊆ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
121 iunss2 4501 . . . . . . . . . . . . 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 7513 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥)))
125124anim1d 586 . . . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝐵 +𝑜 𝑦) → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
127126eleq2d 2673 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +𝑜 𝑦) → (𝑤 ∈ (𝐴 +𝑜 𝑧) ↔ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))))
128127rspcev 3282 . . . . . . . . . . . . . . . . . 18 (((𝐵 +𝑜 𝑦) ∈ (𝐵 +𝑜 𝑥) ∧ 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦))) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))
129125, 128syl6 34 . . . . . . . . . . . . . . . . 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 3012 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦𝑥 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧)))
132 eliun 4460 . . . . . . . . . . . . . . 15 (𝑤 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) ↔ ∃𝑦𝑥 𝑤 ∈ (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
133 eliun 4460 . . . . . . . . . . . . . . 15 (𝑤 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) ↔ ∃𝑧 ∈ (𝐵 +𝑜 𝑥)𝑤 ∈ (𝐴 +𝑜 𝑧))
134131, 132, 1333imtr4g 284 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)) → 𝑤 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧)))
135134ssrdv 3574 . . . . . . . . . . . . 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 3585 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +𝑜 𝑥)(𝐴 +𝑜 𝑧) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
13950, 138eqtrd 2644 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +𝑜 (𝐵 +𝑜 𝑥)) = 𝑦𝑥 (𝐴 +𝑜 (𝐵 +𝑜 𝑦)))
140139an12s 839 . . . . . . . 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 4473 . . . . . . . 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 2647 . . . . . 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 2647 . . . . 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 628 . . . 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 6956 . . 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 1253 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +𝑜 coa 7444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451 This theorem is referenced by:  odi  7546  oaabs  7611  oaabs2  7612
 Copyright terms: Public domain W3C validator