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Theorem oaass 8597
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 7438 . . . . 5 (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
2 oveq2 7438 . . . . . 6 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
32oveq2d 7446 . . . . 5 (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
41, 3eqeq12d 2750 . . . 4 (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
5 oveq2 7438 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
6 oveq2 7438 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7446 . . . . 5 (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
85, 7eqeq12d 2750 . . . 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 7438 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
10 oveq2 7438 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1110oveq2d 7446 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
129, 11eqeq12d 2750 . . . 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 7438 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
14 oveq2 7438 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1514oveq2d 7446 . . . . 5 (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
1613, 15eqeq12d 2750 . . . 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 8571 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
18 oa0 8552 . . . . . 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 8552 . . . . . . 7 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
2120oveq2d 7446 . . . . . 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 2777 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅)))
24 suceq 6451 . . . . . 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 8560 . . . . . . . 8 (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
2617, 25sylan 580 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27 oasuc 8560 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
2827oveq2d 7446 . . . . . . . . . 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 8571 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
31 oasuc 8560 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3230, 31sylan2 593 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3329, 32eqtrd 2774 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3433anassrs 467 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3526, 34eqeq12d 2750 . . . . . 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 246 . . . . 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 413 . . . 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 5015 . . . . . . 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 481 . . . . . 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 3481 . . . . . . . . . 10 𝑥 ∈ V
41 oalim 8568 . . . . . . . . . 10 (((𝐴 +o 𝐵) ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4240, 41mpanr1 703 . . . . . . . . 9 (((𝐴 +o 𝐵) ∈ On ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4317, 42sylan 580 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4443ancoms 458 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4544adantr 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))
46 oalimcl 8596 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
4740, 46mpanr1 703 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
4847ancoms 458 . . . . . . . . . 10 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
49 ovex 7463 . . . . . . . . . . 11 (𝐵 +o 𝑥) ∈ V
50 oalim 8568 . . . . . . . . . . 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 703 . . . . . . . . . 10 ((𝐴 ∈ On ∧ Lim (𝐵 +o 𝑥)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
5248, 51sylan2 593 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
53 limelon 6449 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5440, 53mpan 690 . . . . . . . . . . . . . . . 16 (Lim 𝑥𝑥 ∈ On)
55 oacl 8571 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
5655ancoms 458 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
57 onelon 6410 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
5857ex 412 . . . . . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
6160adantl 481 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
62 0ellim 6448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑥 → ∅ ∈ 𝑥)
63 onelss 6427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧𝐵𝑧𝐵))
6420sseq2d 4027 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +o ∅) ↔ 𝑧𝐵))
6563, 64sylibrd 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵 ∈ On → (𝑧𝐵𝑧 ⊆ (𝐵 +o ∅)))
6665imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ On ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐵 +o ∅))
67 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → (𝐵 +o 𝑦) = (𝐵 +o ∅))
6867sseq2d 4027 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ 𝑧 ⊆ (𝐵 +o ∅)))
6968rspcev 3621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∅ ∈ 𝑥𝑧 ⊆ (𝐵 +o ∅)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
7062, 66, 69syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧𝐵)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
7170expr 456 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Lim 𝑥𝐵 ∈ On) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
7271adantrl 716 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
7372adantrr 717 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
74 oawordex 8593 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
7574ad2ant2l 746 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 ↔ ∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧))
76 oaord 8583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
77763expb 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
78 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐵 +o 𝑦) = 𝑧 → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
7977, 78sylan9bb 509 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦𝑥𝑧 ∈ (𝐵 +o 𝑥)))
8079an32s 652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥𝑧 ∈ (𝐵 +o 𝑥)))
8180biimpar 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑦𝑥)
82 eqimss2 4054 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝐵 +o 𝑦) = 𝑧𝑧 ⊆ (𝐵 +o 𝑦))
8382ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ⊆ (𝐵 +o 𝑦))
8481, 83jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦)))
8584anasss 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥))) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦)))
8685expcom 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦))))
8786reximdv2 3161 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
8887adantrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (∃𝑦 ∈ On (𝐵 +o 𝑦) = 𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
8975, 88sylbid 240 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
9089adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
91 eloni 6395 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ On → Ord 𝑧)
92 eloni 6395 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → Ord 𝐵)
93 ordtri2or 6483 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧𝐵𝐵𝑧))
9491, 92, 93syl2anr 597 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧𝐵𝐵𝑧))
9594ad2ant2l 746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝑧𝐵𝐵𝑧))
9695adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵𝐵𝑧))
9773, 90, 96mpjaod 860 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
9897exp45 438 . . . . . . . . . . . . . . . . . . . . . . 23 (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))))
9998imp 406 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
10099adantld 490 . . . . . . . . . . . . . . . . . . . . 21 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
101100imp32 418 . . . . . . . . . . . . . . . . . . . 20 (((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
102 simplrr 778 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝑧 ∈ On)
103 onelon 6410 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
104103, 30sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵 +o 𝑦) ∈ On)
105104exp32 420 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (𝑥 ∈ On → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ On)))
106105com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ On → (𝐵 ∈ On → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ On)))
107106imp31 417 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦𝑥) → (𝐵 +o 𝑦) ∈ On)
108107ad4ant24 754 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → (𝐵 +o 𝑦) ∈ On)
109 simpll 767 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On) → 𝐴 ∈ On)
110109ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . 22 ((((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) ∧ 𝑧 ∈ On)) ∧ 𝑦𝑥) → 𝐴 ∈ On)
111 oaword 8585 . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . 14 (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
120119imp31 417 . . . . . . . . . . . . 13 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
121120ralrimiv 3142 . . . . . . . . . . . 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 458 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
125 oaordi 8582 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
126125anim1d 611 . . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +o 𝑦) → (𝐴 +o 𝑧) = (𝐴 +o (𝐵 +o 𝑦)))
128127eleq2d 2824 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐵 +o 𝑦) → (𝑤 ∈ (𝐴 +o 𝑧) ↔ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))))
129128rspcev 3621 . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))))
132131rexlimdv 3150 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
133 eliun 4999 . . . . . . . . . . . . . 14 (𝑤 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ↔ ∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))
134 eliun 4999 . . . . . . . . . . . . . 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 4000 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
13754, 136sylan 580 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
138137adantl 481 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
139124, 138eqssd 4012 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
14052, 139eqtrd 2774 . . . . . . . 8 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
141140an12s 649 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
142141adantr 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)))
14339, 45, 1423eqtr4d 2784 . . . . 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 419 . . . 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 7885 . . 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 1116 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338   ciun 4995  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430   +o coa 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508
This theorem is referenced by:  odi  8615  oaabs  8684  oaabs2  8685  oaabsb  43283  omabs2  43321  ofoaass  43349  naddwordnexlem3  43388  naddwordnexlem4  43390
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