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Theorem omabs 7672
 Description: Ordinal multiplication is also absorbed by powers of ω. (Contributed by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
omabs (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))

Proof of Theorem omabs
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2687 . . . . . . . 8 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2 oveq2 6612 . . . . . . . . . 10 (𝑥 = ∅ → (ω ↑𝑜 𝑥) = (ω ↑𝑜 ∅))
32oveq2d 6620 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 ∅)))
43, 2eqeq12d 2636 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅)))
51, 4imbi12d 334 . . . . . . 7 (𝑥 = ∅ → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅))))
6 eleq2 2687 . . . . . . . 8 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
7 oveq2 6612 . . . . . . . . . 10 (𝑥 = 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝑦))
87oveq2d 6620 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
98, 7eqeq12d 2636 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)))
106, 9imbi12d 334 . . . . . . 7 (𝑥 = 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))))
11 eleq2 2687 . . . . . . . 8 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
12 oveq2 6612 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 suc 𝑦))
1312oveq2d 6620 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)))
1413, 12eqeq12d 2636 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
1511, 14imbi12d 334 . . . . . . 7 (𝑥 = suc 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
16 eleq2 2687 . . . . . . . 8 (𝑥 = 𝐵 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐵))
17 oveq2 6612 . . . . . . . . . 10 (𝑥 = 𝐵 → (ω ↑𝑜 𝑥) = (ω ↑𝑜 𝐵))
1817oveq2d 6620 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (𝐴 ·𝑜 (ω ↑𝑜 𝐵)))
1918, 17eqeq12d 2636 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥) ↔ (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵)))
2016, 19imbi12d 334 . . . . . . 7 (𝑥 = 𝐵 → ((∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)) ↔ (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
21 noel 3895 . . . . . . . . 9 ¬ ∅ ∈ ∅
2221pm2.21i 116 . . . . . . . 8 (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅))
2322a1i 11 . . . . . . 7 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ ∅ → (𝐴 ·𝑜 (ω ↑𝑜 ∅)) = (ω ↑𝑜 ∅)))
24 simprl 793 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ω ∈ On)
25 simpll 789 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ ω)
26 simplr 791 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ∅ ∈ 𝐴)
27 omabslem 7671 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 ω) = ω)
2824, 25, 26, 27syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 ω) = ω)
2928adantr 481 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 ω) = ω)
30 suceq 5749 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → suc 𝑦 = suc ∅)
31 df-1o 7505 . . . . . . . . . . . . . . . . . 18 1𝑜 = suc ∅
3230, 31syl6eqr 2673 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → suc 𝑦 = 1𝑜)
3332oveq2d 6620 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (ω ↑𝑜 suc 𝑦) = (ω ↑𝑜 1𝑜))
34 oe1 7569 . . . . . . . . . . . . . . . . 17 (ω ∈ On → (ω ↑𝑜 1𝑜) = ω)
3534ad2antrl 763 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 1𝑜) = ω)
3633, 35sylan9eqr 2677 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (ω ↑𝑜 suc 𝑦) = ω)
3736oveq2d 6620 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ω))
3829, 37, 363eqtr4d 2665 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))
3938ex 450 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
4039a1dd 50 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
41 oveq1 6611 . . . . . . . . . . . . . 14 ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
42 oesuc 7552 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
4342adantl 482 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 suc 𝑦) = ((ω ↑𝑜 𝑦) ·𝑜 ω))
4443oveq2d 6620 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
45 nnon 7018 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ω → 𝐴 ∈ On)
4645ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ On)
47 oecl 7562 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑𝑜 𝑦) ∈ On)
4847adantl 482 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑𝑜 𝑦) ∈ On)
49 omass 7605 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ (ω ↑𝑜 𝑦) ∈ On ∧ ω ∈ On) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
5046, 48, 24, 49syl3anc 1323 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = (𝐴 ·𝑜 ((ω ↑𝑜 𝑦) ·𝑜 ω)))
5144, 50eqtr4d 2658 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω))
5251, 43eqeq12d 2636 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦) ↔ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) ·𝑜 ω) = ((ω ↑𝑜 𝑦) ·𝑜 ω)))
5341, 52syl5ibr 236 . . . . . . . . . . . . 13 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
5453imim2d 57 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
5554com23 86 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (∅ ∈ 𝑦 → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
56 simprr 795 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝑦 ∈ On)
57 on0eqel 5804 . . . . . . . . . . . 12 (𝑦 ∈ On → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
5856, 57syl 17 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
5940, 55, 58mpjaod 396 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))
6059a1dd 50 . . . . . . . . 9 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
6160anassrs 679 . . . . . . . 8 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ 𝑦 ∈ On) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦))))
6261expcom 451 . . . . . . 7 (𝑦 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ suc 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 suc 𝑦)) = (ω ↑𝑜 suc 𝑦)))))
6345ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
64 simprl 793 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ On)
65 simprr 795 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → Lim 𝑥)
66 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
6765, 66jctil 559 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (𝑥 ∈ V ∧ Lim 𝑥))
68 limelon 5747 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6967, 68syl 17 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝑥 ∈ On)
70 oecl 7562 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ↑𝑜 𝑥) ∈ On)
7164, 69, 70syl2anc 692 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) ∈ On)
7271adantr 481 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
73 1onn 7664 . . . . . . . . . . . . . . . . . 18 1𝑜 ∈ ω
7473a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 1𝑜 ∈ ω)
75 ondif2 7527 . . . . . . . . . . . . . . . . 17 (ω ∈ (On ∖ 2𝑜) ↔ (ω ∈ On ∧ 1𝑜 ∈ ω))
7664, 74, 75sylanbrc 697 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ (On ∖ 2𝑜))
7776adantr 481 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
7867adantr 481 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑥 ∈ V ∧ Lim 𝑥))
79 oelimcl 7625 . . . . . . . . . . . . . . 15 ((ω ∈ (On ∖ 2𝑜) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (ω ↑𝑜 𝑥))
8077, 78, 79syl2anc 692 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → Lim (ω ↑𝑜 𝑥))
81 omlim 7558 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ ((ω ↑𝑜 𝑥) ∈ On ∧ Lim (ω ↑𝑜 𝑥))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧))
8263, 72, 80, 81syl12anc 1321 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧))
83 simplrl 799 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ω ∈ On)
84 oelim2 7620 . . . . . . . . . . . . . . . . . . . 20 ((ω ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (ω ↑𝑜 𝑥) = 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
8583, 78, 84syl2anc 692 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) = 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦))
8685eleq2d 2684 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ 𝑧 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦)))
87 eliun 4490 . . . . . . . . . . . . . . . . . 18 (𝑧 𝑦 ∈ (𝑥 ∖ 1𝑜)(ω ↑𝑜 𝑦) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦))
8886, 87syl6bb 276 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
8969adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
90 anass 680 . . . . . . . . . . . . . . . . . . . 20 (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))))
91 onelon 5707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
92 on0eln0 5739 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (∅ ∈ 𝑦𝑦 ≠ ∅))
9391, 92syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦𝑥) → (∅ ∈ 𝑦𝑦 ≠ ∅))
9493pm5.32da 672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ (𝑦𝑥𝑦 ≠ ∅)))
95 dif1o 7525 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝑥 ∖ 1𝑜) ↔ (𝑦𝑥𝑦 ≠ ∅))
9694, 95syl6bbr 278 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ 𝑦 ∈ (𝑥 ∖ 1𝑜)))
9796anbi1d 740 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ (𝑦 ∈ (𝑥 ∖ 1𝑜) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
9890, 97syl5bbr 274 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → ((𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) ↔ (𝑦 ∈ (𝑥 ∖ 1𝑜) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))))
9998rexbidv2 3041 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
10089, 99syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1𝑜)𝑧 ∈ (ω ↑𝑜 𝑦)))
10188, 100bitr4d 271 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) ↔ ∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))))
102 r19.29 3065 . . . . . . . . . . . . . . . . . 18 ((∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → ∃𝑦𝑥 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))))
103 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)))
104103imp 445 . . . . . . . . . . . . . . . . . . . . . 22 (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∅ ∈ 𝑦) → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))
105104anim1i 591 . . . . . . . . . . . . . . . . . . . . 21 ((((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)))
106105anasss 678 . . . . . . . . . . . . . . . . . . . 20 (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)))
10771ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ∈ On)
108 eloni 5692 . . . . . . . . . . . . . . . . . . . . . . 23 ((ω ↑𝑜 𝑥) ∈ On → Ord (ω ↑𝑜 𝑥))
109107, 108syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → Ord (ω ↑𝑜 𝑥))
110 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ (ω ↑𝑜 𝑦))
11164ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ On)
11269ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑥 ∈ On)
113 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑦𝑥)
114112, 113, 91syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑦 ∈ On)
115111, 114, 47syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑦) ∈ On)
116 onelon 5707 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((ω ↑𝑜 𝑦) ∈ On ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → 𝑧 ∈ On)
117115, 110, 116syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝑧 ∈ On)
11845ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝐴 ∈ On)
119118ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → 𝐴 ∈ On)
120 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ∅ ∈ 𝐴)
121120ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
122 omord2 7592 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ On ∧ (ω ↑𝑜 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ω ↑𝑜 𝑦) ↔ (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦))))
123117, 115, 119, 121, 122syl31anc 1326 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑦) ↔ (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦))))
124110, 123mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (𝐴 ·𝑜 (ω ↑𝑜 𝑦)))
125 simprl 793 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))
126124, 125eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦))
12776ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → ω ∈ (On ∖ 2𝑜))
128 oeord 7613 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ ω ∈ (On ∖ 2𝑜)) → (𝑦𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
129114, 112, 127, 128syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝑦𝑥 ↔ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)))
130113, 129mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥))
131 ontr1 5730 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ω ↑𝑜 𝑥) ∈ On → (((𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦) ∧ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)))
132107, 131syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (((𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑦) ∧ (ω ↑𝑜 𝑦) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)))
133126, 130, 132mp2and 714 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥))
134 ordelss 5698 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord (ω ↑𝑜 𝑥) ∧ (𝐴 ·𝑜 𝑧) ∈ (ω ↑𝑜 𝑥)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
135109, 133, 134syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
136135ex 450 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) → (((𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦) ∧ 𝑧 ∈ (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
137106, 136syl5 34 . . . . . . . . . . . . . . . . . . 19 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) → (((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
138137rexlimdva 3024 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∃𝑦𝑥 ((∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
139102, 138syl5 34 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ((∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) ∧ ∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
140139expdimp 453 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
141101, 140sylbid 230 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝑧 ∈ (ω ↑𝑜 𝑥) → (𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥)))
142141ralrimiv 2959 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∀𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
143 iunss 4527 . . . . . . . . . . . . . 14 ( 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥) ↔ ∀𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
144142, 143sylibr 224 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → 𝑧 ∈ (ω ↑𝑜 𝑥)(𝐴 ·𝑜 𝑧) ⊆ (ω ↑𝑜 𝑥))
14582, 144eqsstrd 3618 . . . . . . . . . . . 12 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) ⊆ (ω ↑𝑜 𝑥))
146 simpllr 798 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → ∅ ∈ 𝐴)
147 omword2 7599 . . . . . . . . . . . . 13 ((((ω ↑𝑜 𝑥) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
14872, 63, 146, 147syl21anc 1322 . . . . . . . . . . . 12 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (ω ↑𝑜 𝑥) ⊆ (𝐴 ·𝑜 (ω ↑𝑜 𝑥)))
149145, 148eqssd 3600 . . . . . . . . . . 11 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦))) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥))
150149ex 450 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))
151150anassrs 679 . . . . . . . . 9 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))
152151a1dd 50 . . . . . . . 8 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥))))
153152expcom 451 . . . . . . 7 (Lim 𝑥 → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·𝑜 (ω ↑𝑜 𝑦)) = (ω ↑𝑜 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·𝑜 (ω ↑𝑜 𝑥)) = (ω ↑𝑜 𝑥)))))
1545, 10, 15, 20, 23, 62, 153tfinds3 7011 . . . . . 6 (𝐵 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
155154com12 32 . . . . 5 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
156155adantrr 752 . . . 4 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))))
157156imp32 449 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
158157an32s 845 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
159 nnm0 7630 . . . 4 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
160159ad3antrrr 765 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 ∅) = ∅)
161 fnoe 7535 . . . . . . 7 𝑜 Fn (On × On)
162 fndm 5948 . . . . . . 7 ( ↑𝑜 Fn (On × On) → dom ↑𝑜 = (On × On))
163161, 162ax-mp 5 . . . . . 6 dom ↑𝑜 = (On × On)
164163ndmov 6771 . . . . 5 (¬ (ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑𝑜 𝐵) = ∅)
165164adantl 482 . . . 4 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (ω ↑𝑜 𝐵) = ∅)
166165oveq2d 6620 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (𝐴 ·𝑜 ∅))
167160, 166, 1653eqtr4d 2665 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
168158, 167pm2.61dan 831 1 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 (ω ↑𝑜 𝐵)) = (ω ↑𝑜 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  Vcvv 3186   ∖ cdif 3552   ⊆ wss 3555  ∅c0 3891  ∪ ciun 4485   × cxp 5072  dom cdm 5074  Ord word 5681  Oncon0 5682  Lim wlim 5683  suc csuc 5684   Fn wfn 5842  (class class class)co 6604  ωcom 7012  1𝑜c1o 7498  2𝑜c2o 7499   ·𝑜 comu 7503   ↑𝑜 coe 7504 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511 This theorem is referenced by:  cnfcom3  8545
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