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Theorem oaass 8561
Description: Ordinal addition is associative. Theorem 25 of [Suppes] p. 211. Theorem 4.2 of [Schloeder] p. 11. (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 7417 . . . . 5 (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
2 oveq2 7417 . . . . . 6 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
32oveq2d 7425 . . . . 5 (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
41, 3eqeq12d 2749 . . . 4 (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
5 oveq2 7417 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
6 oveq2 7417 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7425 . . . . 5 (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
85, 7eqeq12d 2749 . . . 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 7417 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
10 oveq2 7417 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1110oveq2d 7425 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
129, 11eqeq12d 2749 . . . 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 7417 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
14 oveq2 7417 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1514oveq2d 7425 . . . . 5 (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
1613, 15eqeq12d 2749 . . . 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 8535 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
18 oa0 8516 . . . . . 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 8516 . . . . . . 7 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
2120oveq2d 7425 . . . . . 6 (𝐵 ∈ On → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
2221adantl 483 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
2319, 22eqtr4d 2776 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅)))
24 suceq 6431 . . . . . 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 8524 . . . . . . . 8 (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
2617, 25sylan 581 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27 oasuc 8524 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
2827oveq2d 7425 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
2928adantl 483 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
30 oacl 8535 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
31 oasuc 8524 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3230, 31sylan2 594 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3329, 32eqtrd 2773 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3433anassrs 469 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3526, 34eqeq12d 2749 . . . . . 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, 35imbitrrid 245 . . . . 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 415 . . . 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 iuneq2 5017 . . . . . . 7 (∀𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦) = (𝐴 +o (𝐵 +o 𝑦)) → 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
3938adantl 483 . . . . . 6 (((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)))
40 vex 3479 . . . . . . . . . 10 𝑥 ∈ V
41 oalim 8532 . . . . . . . . . 10 (((𝐴 +o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4240, 41mpanr1 702 . . . . . . . . 9 (((𝐴 +o 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4317, 42sylan 581 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4443ancoms 460 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4544adantr 482 . . . . . 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))
46 oalimcl 8560 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
4740, 46mpanr1 702 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
4847ancoms 460 . . . . . . . . . 10 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
49 ovex 7442 . . . . . . . . . . 11 (𝐵 +o 𝑥) ∈ V
50 oalim 8532 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ ((𝐵 +o 𝑥) ∈ V ∧ Lim (𝐵 +o 𝑥))) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
5149, 50mpanr1 702 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
5248, 51sylan2 594 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
53 limelon 6429 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5440, 53mpan 689 . . . . . . . . . . . . . . . 16 (Lim 𝑥𝑥 ∈ On)
55 oacl 8535 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
5655ancoms 460 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
57 onelon 6390 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
5857ex 414 . . . . . . . . . . . . . . . . . . . . 21 ((𝐵 +o 𝑥) ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → 𝑧 ∈ On))
5956, 58syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → 𝑧 ∈ On))
6059adantld 492 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
6160adantl 483 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
62 0ellim 6428 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑥 → ∅ ∈ 𝑥)
63 onelss 6407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧𝐵𝑧𝐵))
6420sseq2d 4015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +o ∅) ↔ 𝑧𝐵))
6563, 64sylibrd 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵 ∈ On → (𝑧𝐵𝑧 ⊆ (𝐵 +o ∅)))
6665imp 408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ On ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐵 +o ∅))
67 oveq2 7417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → (𝐵 +o 𝑦) = (𝐵 +o ∅))
6867sseq2d 4015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ 𝑧 ⊆ (𝐵 +o ∅)))
6968rspcev 3613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∅ ∈ 𝑥𝑧 ⊆ (𝐵 +o ∅)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
7062, 66, 69syl2an 597 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧𝐵)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
7170expr 458 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Lim 𝑥𝐵 ∈ On) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
7271adantrl 715 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
7372adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
74 oawordex 8557 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
7574ad2ant2l 745 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
76 oaord 8547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
77763expb 1121 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
78 eleq1 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐵 +o 𝑦) = 𝑧 → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
7977, 78sylan9bb 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦𝑥𝑧 ∈ (𝐵 +o 𝑥)))
8079an32s 651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥𝑧 ∈ (𝐵 +o 𝑥)))
8180biimpar 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑦𝑥)
82 eqimss2 4042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐵 +o 𝑦) = 𝑧𝑧 ⊆ (𝐵 +o 𝑦))
8382ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ⊆ (𝐵 +o 𝑦))
8481, 83jca 513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦)))
8584anasss 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥))) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦)))
8685expcom 415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦))))
8786reximdv2 3165 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
8887adantrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
8975, 88sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
9089adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
91 eloni 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ On → Ord 𝑧)
92 eloni 6375 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → Ord 𝐵)
93 ordtri2or 6463 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧𝐵𝐵𝑧))
9491, 92, 93syl2anr 598 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧𝐵𝐵𝑧))
9594ad2ant2l 745 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝑧𝐵𝐵𝑧))
9695adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵𝐵𝑧))
9773, 90, 96mpjaod 859 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
9897exp45 440 . . . . . . . . . . . . . . . . . . . . . . 23 (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))))
9998imp 408 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
10099adantld 492 . . . . . . . . . . . . . . . . . . . . 21 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
101100imp32 420 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
102 simplrr 777 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝑧 ∈ On)
103 onelon 6390 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
104103, 30sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵 +o 𝑦) ∈ On)
105104exp32 422 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (𝑥 ∈ On → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ On)))
106105com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ On → (𝐵 ∈ On → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ On)))
107106imp31 419 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦𝑥) → (𝐵 +o 𝑦) ∈ On)
108107ad4ant24 753 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → (𝐵 +o 𝑦) ∈ On)
109 simpll 766 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
110109ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
111 oaword 8549 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ On ∧ (𝐵 +o 𝑦) ∈ On ∧ 𝐴 ∈ On) → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
112102, 108, 110, 111syl3anc 1372 . . . . . . . . . . . . . . . . . . . . 21 ((((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))))
113112rexbidva 3177 . . . . . . . . . . . . . . . . . . . 20 (((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))))
114101, 113mpbid 231 . . . . . . . . . . . . . . . . . . 19 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))
115114exp32 422 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))))
11661, 115mpdd 43 . . . . . . . . . . . . . . . . 17 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
117116exp32 422 . . . . . . . . . . . . . . . 16 (Lim 𝑥 → (𝑥 ∈ On → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
11854, 117mpd 15 . . . . . . . . . . . . . . 15 (Lim 𝑥 → (𝐵 ∈ On → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))))
119118exp4a 433 . . . . . . . . . . . . . 14 (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
120119imp31 419 . . . . . . . . . . . . 13 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
121120ralrimiv 3146 . . . . . . . . . . . 12 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))
122 iunss2 5053 . . . . . . . . . . . 12 (∀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)))
123121, 122syl 17 . . . . . . . . . . 11 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
124123ancoms 460 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
125 oaordi 8546 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
126125anim1d 612 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦𝑥𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))) → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ∧ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))))
127 oveq2 7417 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +o 𝑦) → (𝐴 +o 𝑧) = (𝐴 +o (𝐵 +o 𝑦)))
128127eleq2d 2820 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐵 +o 𝑦) → (𝑤 ∈ (𝐴 +o 𝑧) ↔ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))))
129128rspcev 3613 . . . . . . . . . . . . . . . . 17 (((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ∧ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))
130126, 129syl6 35 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝑦𝑥𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
131130expd 417 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))))
132131rexlimdv 3154 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
133 eliun 5002 . . . . . . . . . . . . . 14 (𝑤 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ↔ ∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))
134 eliun 5002 . . . . . . . . . . . . . 14 (𝑤 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ↔ ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))
135132, 133, 1343imtr4g 296 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) → 𝑤 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧)))
136135ssrdv 3989 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
13754, 136sylan 581 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
138137adantl 483 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
139124, 138eqssd 4000 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
14052, 139eqtrd 2773 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
141140an12s 648 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
142141adantr 482 . . . . . 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)))
14339, 45, 1423eqtr4d 2783 . . . . 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)))
144143exp31 421 . . . 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)))))
1454, 8, 12, 16, 23, 37, 144tfinds3 7854 . . 3 (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
146145com12 32 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))))
1471463impia 1118 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 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  wss 3949  c0 4323   ciun 4998  Ord word 6364  Oncon0 6365  Lim wlim 6366  suc csuc 6367  (class class class)co 7409   +o coa 8463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-oadd 8470
This theorem is referenced by:  odi  8579  oaabs  8647  oaabs2  8648  oaabsb  42044  omabs2  42082  ofoaass  42110  naddwordnexlem3  42150  naddwordnexlem4  42152
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