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Theorem oaass 8512
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 7369 . . . . 5 (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
2 oveq2 7369 . . . . . 6 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
32oveq2d 7377 . . . . 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 7369 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
6 oveq2 7369 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7377 . . . . 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 7369 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
10 oveq2 7369 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1110oveq2d 7377 . . . . 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 7369 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
14 oveq2 7369 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1514oveq2d 7377 . . . . 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 8485 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
18 oa0 8466 . . . . . 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 8466 . . . . . . 7 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
2120oveq2d 7377 . . . . . 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 6387 . . . . . 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 8474 . . . . . . . 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 8474 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
2827oveq2d 7377 . . . . . . . . . 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 8485 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
31 oasuc 8474 . . . . . . . . . 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 4977 . . . . . . 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 3451 . . . . . . . . . 10 𝑥 ∈ V
41 oalim 8482 . . . . . . . . . 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 8511 . . . . . . . . . . . 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 7394 . . . . . . . . . . 11 (𝐵 +o 𝑥) ∈ V
50 oalim 8482 . . . . . . . . . . 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 6385 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5440, 53mpan 689 . . . . . . . . . . . . . . . 16 (Lim 𝑥𝑥 ∈ On)
55 oacl 8485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
5655ancoms 460 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
57 onelon 6346 . . . . . . . . . . . . . . . . . . . . . 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 6384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑥 → ∅ ∈ 𝑥)
63 onelss 6363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧𝐵𝑧𝐵))
6420sseq2d 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → (𝐵 +o 𝑦) = (𝐵 +o ∅))
6867sseq2d 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ 𝑧 ⊆ (𝐵 +o ∅)))
6968rspcev 3583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8508 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3158 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6331 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ On → Ord 𝑧)
92 eloni 6331 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → Ord 𝐵)
93 ordtri2or 6419 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6346 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8500 . . . . . . . . . . . . . . . . . . . . . 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 3170 . . . . . . . . . . . . . . . . . . . 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 3139 . . . . . . . . . . . 12 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))
122 iunss2 5013 . . . . . . . . . . . 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 8497 . . . . . . . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . . . . . . 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 3583 . . . . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
133 eliun 4962 . . . . . . . . . . . . . 14 (𝑤 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ↔ ∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))
134 eliun 4962 . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . 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 3965 . . . . . . . . 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 7805 . . 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 3061  wrex 3070  Vcvv 3447  wss 3914  c0 4286   ciun 4958  Ord word 6320  Oncon0 6321  Lim wlim 6322  suc csuc 6323  (class class class)co 7361   +o coa 8413
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 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-oadd 8420
This theorem is referenced by:  odi  8530  oaabs  8598  oaabs2  8599  oaabsb  41676  omabs2  41714  ofoaass  41723  naddwordnexlem3  41763  naddwordnexlem4  41765
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