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Theorem omabs 8011
 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 2848 . . . . . . . 8 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2 oveq2 6930 . . . . . . . . . 10 (𝑥 = ∅ → (ω ↑o 𝑥) = (ω ↑o ∅))
32oveq2d 6938 . . . . . . . . 9 (𝑥 = ∅ → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o ∅)))
43, 2eqeq12d 2793 . . . . . . . 8 (𝑥 = ∅ → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅)))
51, 4imbi12d 336 . . . . . . 7 (𝑥 = ∅ → ((∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)) ↔ (∅ ∈ ∅ → (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅))))
6 eleq2 2848 . . . . . . . 8 (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
7 oveq2 6930 . . . . . . . . . 10 (𝑥 = 𝑦 → (ω ↑o 𝑥) = (ω ↑o 𝑦))
87oveq2d 6938 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o 𝑦)))
98, 7eqeq12d 2793 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)))
106, 9imbi12d 336 . . . . . . 7 (𝑥 = 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)) ↔ (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))))
11 eleq2 2848 . . . . . . . 8 (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
12 oveq2 6930 . . . . . . . . . 10 (𝑥 = suc 𝑦 → (ω ↑o 𝑥) = (ω ↑o suc 𝑦))
1312oveq2d 6938 . . . . . . . . 9 (𝑥 = suc 𝑦 → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o suc 𝑦)))
1413, 12eqeq12d 2793 . . . . . . . 8 (𝑥 = suc 𝑦 → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o suc 𝑦)) = (ω ↑o suc 𝑦)))
1511, 14imbi12d 336 . . . . . . 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 2848 . . . . . . . 8 (𝑥 = 𝐵 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐵))
17 oveq2 6930 . . . . . . . . . 10 (𝑥 = 𝐵 → (ω ↑o 𝑥) = (ω ↑o 𝐵))
1817oveq2d 6938 . . . . . . . . 9 (𝑥 = 𝐵 → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o 𝐵)))
1918, 17eqeq12d 2793 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵)))
2016, 19imbi12d 336 . . . . . . 7 (𝑥 = 𝐵 → ((∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)) ↔ (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
21 noel 4146 . . . . . . . . 9 ¬ ∅ ∈ ∅
2221pm2.21i 117 . . . . . . . 8 (∅ ∈ ∅ → (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅))
2322a1i 11 . . . . . . 7 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ ∅ → (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅)))
24 simprl 761 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ω ∈ On)
25 simpll 757 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ ω)
26 simplr 759 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ∅ ∈ 𝐴)
27 omabslem 8010 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
2824, 25, 26, 27syl3anc 1439 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o ω) = ω)
2928adantr 474 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·o ω) = ω)
30 suceq 6041 . . . . . . . . . . . . . . . . . 18 (𝑦 = ∅ → suc 𝑦 = suc ∅)
31 df-1o 7843 . . . . . . . . . . . . . . . . . 18 1o = suc ∅
3230, 31syl6eqr 2832 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → suc 𝑦 = 1o)
3332oveq2d 6938 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (ω ↑o suc 𝑦) = (ω ↑o 1o))
34 oe1 7908 . . . . . . . . . . . . . . . . 17 (ω ∈ On → (ω ↑o 1o) = ω)
3534ad2antrl 718 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑o 1o) = ω)
3633, 35sylan9eqr 2836 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (ω ↑o suc 𝑦) = ω)
3736oveq2d 6938 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·o (ω ↑o suc 𝑦)) = (𝐴 ·o ω))
3829, 37, 363eqtr4d 2824 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·o (ω ↑o suc 𝑦)) = (ω ↑o suc 𝑦))
3938ex 403 . . . . . . . . . . . 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 6929 . . . . . . . . . . . . . 14 ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) → ((𝐴 ·o (ω ↑o 𝑦)) ·o ω) = ((ω ↑o 𝑦) ·o ω))
42 oesuc 7891 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑o suc 𝑦) = ((ω ↑o 𝑦) ·o ω))
4342adantl 475 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑o suc 𝑦) = ((ω ↑o 𝑦) ·o ω))
4443oveq2d 6938 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o (ω ↑o suc 𝑦)) = (𝐴 ·o ((ω ↑o 𝑦) ·o ω)))
45 nnon 7349 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ω → 𝐴 ∈ On)
4645ad2antrr 716 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ On)
47 oecl 7901 . . . . . . . . . . . . . . . . . 18 ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑o 𝑦) ∈ On)
4847adantl 475 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑o 𝑦) ∈ On)
49 omass 7944 . . . . . . . . . . . . . . . . 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 1439 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·o (ω ↑o 𝑦)) ·o ω) = (𝐴 ·o ((ω ↑o 𝑦) ·o ω)))
5144, 50eqtr4d 2817 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o (ω ↑o suc 𝑦)) = ((𝐴 ·o (ω ↑o 𝑦)) ·o ω))
5251, 43eqeq12d 2793 . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . 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 763 . . . . . . . . . . . 12 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝑦 ∈ On)
57 on0eqel 6093 . . . . . . . . . . . 12 (𝑦 ∈ On → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
5856, 57syl 17 . . . . . . . . . . 11 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
5940, 55, 58mpjaod 849 . . . . . . . . . 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 461 . . . . . . . 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 404 . . . . . . 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 720 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → 𝐴 ∈ On)
64 simprl 761 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ On)
65 simprr 763 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → Lim 𝑥)
66 vex 3401 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
6765, 66jctil 515 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (𝑥 ∈ V ∧ Lim 𝑥))
68 limelon 6039 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
6967, 68syl 17 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝑥 ∈ On)
70 oecl 7901 . . . . . . . . . . . . . . . 16 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ↑o 𝑥) ∈ On)
7164, 69, 70syl2anc 579 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (ω ↑o 𝑥) ∈ On)
7271adantr 474 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (ω ↑o 𝑥) ∈ On)
73 1onn 8003 . . . . . . . . . . . . . . . . 17 1o ∈ ω
74 ondif2 7866 . . . . . . . . . . . . . . . . 17 (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
7564, 73, 74sylanblrc 584 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ (On ∖ 2o))
7675adantr 474 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ω ∈ (On ∖ 2o))
7767adantr 474 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝑥 ∈ V ∧ Lim 𝑥))
78 oelimcl 7964 . . . . . . . . . . . . . . 15 ((ω ∈ (On ∖ 2o) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (ω ↑o 𝑥))
7976, 77, 78syl2anc 579 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → Lim (ω ↑o 𝑥))
80 omlim 7897 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ ((ω ↑o 𝑥) ∈ On ∧ Lim (ω ↑o 𝑥))) → (𝐴 ·o (ω ↑o 𝑥)) = 𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧))
8163, 72, 79, 80syl12anc 827 . . . . . . . . . . . . 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))
82 simplrl 767 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ω ∈ On)
83 oelim2 7959 . . . . . . . . . . . . . . . . . . . 20 ((ω ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (ω ↑o 𝑥) = 𝑦 ∈ (𝑥 ∖ 1o)(ω ↑o 𝑦))
8482, 77, 83syl2anc 579 . . . . . . . . . . . . . . . . . . 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))
8584eleq2d 2845 . . . . . . . . . . . . . . . . . 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)))
86 eliun 4757 . . . . . . . . . . . . . . . . . 18 (𝑧 𝑦 ∈ (𝑥 ∖ 1o)(ω ↑o 𝑦) ↔ ∃𝑦 ∈ (𝑥 ∖ 1o)𝑧 ∈ (ω ↑o 𝑦))
8785, 86syl6bb 279 . . . . . . . . . . . . . . . . 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)))
8869adantr 474 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → 𝑥 ∈ On)
89 anass 462 . . . . . . . . . . . . . . . . . . . 20 (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦)) ↔ (𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑o 𝑦))))
90 onelon 6001 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
91 on0eln0 6031 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (∅ ∈ 𝑦𝑦 ≠ ∅))
9290, 91syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦𝑥) → (∅ ∈ 𝑦𝑦 ≠ ∅))
9392pm5.32da 574 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ (𝑦𝑥𝑦 ≠ ∅)))
94 dif1o 7864 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (𝑥 ∖ 1o) ↔ (𝑦𝑥𝑦 ≠ ∅))
9593, 94syl6bbr 281 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ On → ((𝑦𝑥 ∧ ∅ ∈ 𝑦) ↔ 𝑦 ∈ (𝑥 ∖ 1o)))
9695anbi1d 623 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ On → (((𝑦𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦)) ↔ (𝑦 ∈ (𝑥 ∖ 1o) ∧ 𝑧 ∈ (ω ↑o 𝑦))))
9789, 96syl5bbr 277 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On → ((𝑦𝑥 ∧ (∅ ∈ 𝑦𝑧 ∈ (ω ↑o 𝑦))) ↔ (𝑦 ∈ (𝑥 ∖ 1o) ∧ 𝑧 ∈ (ω ↑o 𝑦))))
9897rexbidv2 3233 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → (∃𝑦𝑥 (∅ ∈ 𝑦𝑧 ∈ (ω ↑o 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1o)𝑧 ∈ (ω ↑o 𝑦)))
9988, 98syl 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)))
10087, 99bitr4d 274 . . . . . . . . . . . . . . . 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))))
101 r19.29 3258 . . . . . . . . . . . . . . . . . 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))))
102 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 ((∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) → (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)))
103102imp 397 . . . . . . . . . . . . . . . . . . . . . 22 (((∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ ∅ ∈ 𝑦) → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))
104103anim1i 608 . . . . . . . . . . . . . . . . . . . . 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)))
105104anasss 460 . . . . . . . . . . . . . . . . . . . 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)))
10671ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . 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)
107 eloni 5986 . . . . . . . . . . . . . . . . . . . . . . 23 ((ω ↑o 𝑥) ∈ On → Ord (ω ↑o 𝑥))
108106, 107syl 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))
109 simprr 763 . . . . . . . . . . . . . . . . . . . . . . . . 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))
11064ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ω ∈ On)
11169ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑥 ∈ On)
112 simplr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑦𝑥)
113111, 112, 90syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑦 ∈ On)
114110, 113, 47syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
115 onelon 6001 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((ω ↑o 𝑦) ∈ On ∧ 𝑧 ∈ (ω ↑o 𝑦)) → 𝑧 ∈ On)
116114, 109, 115syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑧 ∈ On)
11745ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝐴 ∈ On)
118117ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝐴 ∈ On)
119 simplr 759 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ∅ ∈ 𝐴)
120119ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ∅ ∈ 𝐴)
121 omord2 7931 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑧 ∈ On ∧ (ω ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ω ↑o 𝑦) ↔ (𝐴 ·o 𝑧) ∈ (𝐴 ·o (ω ↑o 𝑦))))
122116, 114, 118, 120, 121syl31anc 1441 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
123109, 122mpbid 224 . . . . . . . . . . . . . . . . . . . . . . . 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)))
124 simprl 761 . . . . . . . . . . . . . . . . . . . . . . . 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))
125123, 124eleqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . 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))
12675ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ω ∈ (On ∖ 2o))
127 oeord 7952 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ ω ∈ (On ∖ 2o)) → (𝑦𝑥 ↔ (ω ↑o 𝑦) ∈ (ω ↑o 𝑥)))
128113, 111, 126, 127syl3anc 1439 . . . . . . . . . . . . . . . . . . . . . . . 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)))
129112, 128mpbid 224 . . . . . . . . . . . . . . . . . . . . . . 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))
130 ontr1 6022 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ω ↑o 𝑥) ∈ On → (((𝐴 ·o 𝑧) ∈ (ω ↑o 𝑦) ∧ (ω ↑o 𝑦) ∈ (ω ↑o 𝑥)) → (𝐴 ·o 𝑧) ∈ (ω ↑o 𝑥)))
131106, 130syl 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)))
132125, 129, 131mp2and 689 . . . . . . . . . . . . . . . . . . . . . 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))
133 ordelss 5992 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord (ω ↑o 𝑥) ∧ (𝐴 ·o 𝑧) ∈ (ω ↑o 𝑥)) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
134108, 132, 133syl2anc 579 . . . . . . . . . . . . . . . . . . . . 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))
135134ex 403 . . . . . . . . . . . . . . . . . . . 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)))
136105, 135syl5 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)))
137136rexlimdva 3213 . . . . . . . . . . . . . . . . . 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)))
138101, 137syl5 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)))
139138expdimp 446 . . . . . . . . . . . . . . . 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)))
140100, 139sylbid 232 . . . . . . . . . . . . . . 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)))
141140ralrimiv 3147 . . . . . . . . . . . . . 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))
142 iunss 4794 . . . . . . . . . . . . . 14 ( 𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥) ↔ ∀𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
143141, 142sylibr 226 . . . . . . . . . . . . 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))
14481, 143eqsstrd 3858 . . . . . . . . . . . 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))
145 simpllr 766 . . . . . . . . . . . . 13 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ∅ ∈ 𝐴)
146 omword2 7938 . . . . . . . . . . . . 13 ((((ω ↑o 𝑥) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (ω ↑o 𝑥) ⊆ (𝐴 ·o (ω ↑o 𝑥)))
14772, 63, 145, 146syl21anc 828 . . . . . . . . . . . 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)))
148144, 147eqssd 3838 . . . . . . . . . . 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))
149148ex 403 . . . . . . . . . 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)))
150149anassrs 461 . . . . . . . . 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)))
151150a1dd 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))))
152151expcom 404 . . . . . . 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)))))
1535, 10, 15, 20, 23, 62, 152tfinds3 7342 . . . . . 6 (𝐵 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
154153com12 32 . . . . 5 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
155154adantrr 707 . . . 4 (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
156155imp32 411 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))
157156an32s 642 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))
158 nnm0 7969 . . . 4 (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
159158ad3antrrr 720 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o ∅) = ∅)
160 fnoe 7874 . . . . . . 7 o Fn (On × On)
161 fndm 6235 . . . . . . 7 ( ↑o Fn (On × On) → dom ↑o = (On × On))
162160, 161ax-mp 5 . . . . . 6 dom ↑o = (On × On)
163162ndmov 7095 . . . . 5 (¬ (ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) = ∅)
164163adantl 475 . . . 4 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (ω ↑o 𝐵) = ∅)
165164oveq2d 6938 . . 3 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (ω ↑o 𝐵)) = (𝐴 ·o ∅))
166159, 165, 1643eqtr4d 2824 . 2 ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))
167157, 166pm2.61dan 803 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 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ∖ cdif 3789   ⊆ wss 3792  ∅c0 4141  ∪ ciun 4753   × cxp 5353  dom cdm 5355  Ord word 5975  Oncon0 5976  Lim wlim 5977  suc csuc 5978   Fn wfn 6130  (class class class)co 6922  ωcom 7343  1oc1o 7836  2oc2o 7837   ·o comu 7841   ↑o coe 7842 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-oexp 7849 This theorem is referenced by:  cnfcom3  8898
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