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Theorem oaass 8174
 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 7148 . . . . 5 (𝑥 = ∅ → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o ∅))
2 oveq2 7148 . . . . . 6 (𝑥 = ∅ → (𝐵 +o 𝑥) = (𝐵 +o ∅))
32oveq2d 7156 . . . . 5 (𝑥 = ∅ → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o ∅)))
41, 3eqeq12d 2838 . . . 4 (𝑥 = ∅ → (((𝐴 +o 𝐵) +o 𝑥) = (𝐴 +o (𝐵 +o 𝑥)) ↔ ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅))))
5 oveq2 7148 . . . . 5 (𝑥 = 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝑦))
6 oveq2 7148 . . . . . 6 (𝑥 = 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o 𝑦))
76oveq2d 7156 . . . . 5 (𝑥 = 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝑦)))
85, 7eqeq12d 2838 . . . 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 7148 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o suc 𝑦))
10 oveq2 7148 . . . . . 6 (𝑥 = suc 𝑦 → (𝐵 +o 𝑥) = (𝐵 +o suc 𝑦))
1110oveq2d 7156 . . . . 5 (𝑥 = suc 𝑦 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o suc 𝑦)))
129, 11eqeq12d 2838 . . . 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 7148 . . . . 5 (𝑥 = 𝐶 → ((𝐴 +o 𝐵) +o 𝑥) = ((𝐴 +o 𝐵) +o 𝐶))
14 oveq2 7148 . . . . . 6 (𝑥 = 𝐶 → (𝐵 +o 𝑥) = (𝐵 +o 𝐶))
1514oveq2d 7156 . . . . 5 (𝑥 = 𝐶 → (𝐴 +o (𝐵 +o 𝑥)) = (𝐴 +o (𝐵 +o 𝐶)))
1613, 15eqeq12d 2838 . . . 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 8147 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ∈ On)
18 oa0 8128 . . . . . 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 8128 . . . . . . 7 (𝐵 ∈ On → (𝐵 +o ∅) = 𝐵)
2120oveq2d 7156 . . . . . 6 (𝐵 ∈ On → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
2221adantl 485 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o (𝐵 +o ∅)) = (𝐴 +o 𝐵))
2319, 22eqtr4d 2860 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) +o ∅) = (𝐴 +o (𝐵 +o ∅)))
24 suceq 6234 . . . . . 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 8136 . . . . . . . 8 (((𝐴 +o 𝐵) ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
2617, 25sylan 583 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → ((𝐴 +o 𝐵) +o suc 𝑦) = suc ((𝐴 +o 𝐵) +o 𝑦))
27 oasuc 8136 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o suc 𝑦) = suc (𝐵 +o 𝑦))
2827oveq2d 7156 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
2928adantl 485 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = (𝐴 +o suc (𝐵 +o 𝑦)))
30 oacl 8147 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑦 ∈ On) → (𝐵 +o 𝑦) ∈ On)
31 oasuc 8136 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (𝐵 +o 𝑦) ∈ On) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3230, 31sylan2 595 . . . . . . . . 9 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o suc (𝐵 +o 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3329, 32eqtrd 2857 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑦 ∈ On)) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3433anassrs 471 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On) → (𝐴 +o (𝐵 +o suc 𝑦)) = suc (𝐴 +o (𝐵 +o 𝑦)))
3526, 34eqeq12d 2838 . . . . . 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 249 . . . . 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 417 . . . 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 4913 . . . . . . 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 485 . . . . . 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 3472 . . . . . . . . . 10 𝑥 ∈ V
41 oalim 8144 . . . . . . . . . 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 583 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ Lim 𝑥) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4443ancoms 462 . . . . . . 7 ((Lim 𝑥 ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 +o 𝐵) +o 𝑥) = 𝑦𝑥 ((𝐴 +o 𝐵) +o 𝑦))
4544adantr 484 . . . . . 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 8173 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (𝐵 +o 𝑥))
4740, 46mpanr1 702 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ Lim 𝑥) → Lim (𝐵 +o 𝑥))
4847ancoms 462 . . . . . . . . . 10 ((Lim 𝑥𝐵 ∈ On) → Lim (𝐵 +o 𝑥))
49 ovex 7173 . . . . . . . . . . 11 (𝐵 +o 𝑥) ∈ V
50 oalim 8144 . . . . . . . . . . 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 595 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → (𝐴 +o (𝐵 +o 𝑥)) = 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
53 limelon 6232 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
5440, 53mpan 689 . . . . . . . . . . . . . . . 16 (Lim 𝑥𝑥 ∈ On)
55 oacl 8147 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐵 ∈ On ∧ 𝑥 ∈ On) → (𝐵 +o 𝑥) ∈ On)
5655ancoms 462 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝑥) ∈ On)
57 onelon 6194 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 +o 𝑥) ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On)
5857ex 416 . . . . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
6160adantl 485 . . . . . . . . . . . . . . . . . 18 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑧 ∈ On))
62 0ellim 6231 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑥 → ∅ ∈ 𝑥)
63 onelss 6211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧𝐵𝑧𝐵))
6420sseq2d 3974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐵 ∈ On → (𝑧 ⊆ (𝐵 +o ∅) ↔ 𝑧𝐵))
6563, 64sylibrd 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝐵 ∈ On → (𝑧𝐵𝑧 ⊆ (𝐵 +o ∅)))
6665imp 410 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐵 ∈ On ∧ 𝑧𝐵) → 𝑧 ⊆ (𝐵 +o ∅))
67 oveq2 7148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → (𝐵 +o 𝑦) = (𝐵 +o ∅))
6867sseq2d 3974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑧 ⊆ (𝐵 +o 𝑦) ↔ 𝑧 ⊆ (𝐵 +o ∅)))
6968rspcev 3598 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∅ ∈ 𝑥𝑧 ⊆ (𝐵 +o ∅)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
7062, 66, 69syl2an 598 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Lim 𝑥 ∧ (𝐵 ∈ On ∧ 𝑧𝐵)) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
7170expr 460 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
77763expb 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑦 ∈ On ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑦𝑥 ↔ (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
78 eleq1 2901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝐵 +o 𝑦) = 𝑧 → ((𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥) ↔ 𝑧 ∈ (𝐵 +o 𝑥)))
7977, 78sylan9bb 513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → 𝑦𝑥)
82 eqimss2 3999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦)))
8584anasss 470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥))) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦)))
8685expcom 417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → ((𝑦 ∈ On ∧ (𝐵 +o 𝑦) = 𝑧) → (𝑦𝑥𝑧 ⊆ (𝐵 +o 𝑦))))
8786reximdv2 3257 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
9089adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝐵𝑧 → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))
91 eloni 6179 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ On → Ord 𝑧)
92 eloni 6179 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → Ord 𝐵)
93 ordtri2or 6264 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Ord 𝑧 ∧ Ord 𝐵) → (𝑧𝐵𝐵𝑧))
9491, 92, 93syl2anr 599 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑧 ∈ On) → (𝑧𝐵𝐵𝑧))
9594ad2ant2l 745 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On)) → (𝑧𝐵𝐵𝑧))
9695adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → (𝑧𝐵𝐵𝑧))
9773, 90, 96mpjaod 857 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Lim 𝑥 ∧ ((𝑥 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑧 ∈ (𝐵 +o 𝑥) ∧ 𝑧 ∈ On))) → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))
9897exp45 442 . . . . . . . . . . . . . . . . . . . . . . 23 (Lim 𝑥 → ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦)))))
9998imp 410 . . . . . . . . . . . . . . . . . . . . . 22 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → (𝑧 ∈ (𝐵 +o 𝑥) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
10099adantld 494 . . . . . . . . . . . . . . . . . . . . 21 ((Lim 𝑥 ∧ (𝑥 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ∈ On ∧ 𝑧 ∈ (𝐵 +o 𝑥)) → (𝑧 ∈ On → ∃𝑦𝑥 𝑧 ⊆ (𝐵 +o 𝑦))))
101100imp32 422 . . . . . . . . . . . . . . . . . . . 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 6194 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
104103, 30sylan2 595 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐵 ∈ On ∧ (𝑥 ∈ On ∧ 𝑦𝑥)) → (𝐵 +o 𝑦) ∈ On)
105104exp32 424 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 ∈ On → (𝑥 ∈ On → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ On)))
106105com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ On → (𝐵 ∈ On → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ On)))
107106imp31 421 . . . . . . . . . . . . . . . . . . . . . . 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 8162 . . . . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . . . . 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 3282 . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . 14 (Lim 𝑥 → (𝐵 ∈ On → (𝐴 ∈ On → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))))
120119imp31 421 . . . . . . . . . . . . 13 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → (𝑧 ∈ (𝐵 +o 𝑥) → ∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦))))
121120ralrimiv 3173 . . . . . . . . . . . 12 (((Lim 𝑥𝐵 ∈ On) ∧ 𝐴 ∈ On) → ∀𝑧 ∈ (𝐵 +o 𝑥)∃𝑦𝑥 (𝐴 +o 𝑧) ⊆ (𝐴 +o (𝐵 +o 𝑦)))
122 iunss2 4948 . . . . . . . . . . . 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 462 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ⊆ 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
125 oaordi 8159 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝐵 +o 𝑦) ∈ (𝐵 +o 𝑥)))
126125anim1d 613 . . . . . . . . . . . . . . . . 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 7148 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐵 +o 𝑦) → (𝐴 +o 𝑧) = (𝐴 +o (𝐵 +o 𝑦)))
128127eleq2d 2899 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝐵 +o 𝑦) → (𝑤 ∈ (𝐴 +o 𝑧) ↔ 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦))))
129128rspcev 3598 . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝑥 → (𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))))
132131rexlimdv 3269 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)) → ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧)))
133 eliun 4898 . . . . . . . . . . . . . 14 (𝑤 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ↔ ∃𝑦𝑥 𝑤 ∈ (𝐴 +o (𝐵 +o 𝑦)))
134 eliun 4898 . . . . . . . . . . . . . 14 (𝑤 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) ↔ ∃𝑧 ∈ (𝐵 +o 𝑥)𝑤 ∈ (𝐴 +o 𝑧))
135132, 133, 1343imtr4g 299 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑤 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) → 𝑤 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧)))
136135ssrdv 3948 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝐵 ∈ On) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
13754, 136sylan 583 . . . . . . . . . . 11 ((Lim 𝑥𝐵 ∈ On) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
138137adantl 485 . . . . . . . . . 10 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)) ⊆ 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧))
139124, 138eqssd 3959 . . . . . . . . 9 ((𝐴 ∈ On ∧ (Lim 𝑥𝐵 ∈ On)) → 𝑧 ∈ (𝐵 +o 𝑥)(𝐴 +o 𝑧) = 𝑦𝑥 (𝐴 +o (𝐵 +o 𝑦)))
14052, 139eqtrd 2857 . . . . . . . 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 484 . . . . . 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 2867 . . . . 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 423 . . . 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 7564 . . 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 1114 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131  Vcvv 3469   ⊆ wss 3908  ∅c0 4265  ∪ ciun 4894  Ord word 6168  Oncon0 6169  Lim wlim 6170  suc csuc 6171  (class class class)co 7140   +o coa 8086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-oadd 8093 This theorem is referenced by:  odi  8192  oaabs  8258  oaabs2  8259
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