Theorem omabs 8605

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.

Step	Hyp	Ref	Expression
1		eleq2 2841	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
2		oveq2 7389	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (ω ↑o 𝑥) = (ω ↑o ∅))
3	2	oveq2d 7397	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o ∅)))
4	3, 2	eqeq12d 2768	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅)))
5	1, 4	imbi12d 346	. . . . . . 7 ⊢ (𝑥 = ∅ → ((∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)) ↔ (∅ ∈ ∅ → (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅))))
6		eleq2 2841	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑦))
7		oveq2 7389	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (ω ↑o 𝑥) = (ω ↑o 𝑦))
8	7	oveq2d 7397	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o 𝑦)))
9	8, 7	eqeq12d 2768	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)))
10	6, 9	imbi12d 346	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)) ↔ (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))))
11		eleq2 2841	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (∅ ∈ 𝑥 ↔ ∅ ∈ suc 𝑦))
12		oveq2 7389	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑦 → (ω ↑o 𝑥) = (ω ↑o suc 𝑦))
13	12	oveq2d 7397	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o suc 𝑦)))
14	13, 12	eqeq12d 2768	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o suc 𝑦)) = (ω ↑o suc 𝑦)))
15	11, 14	imbi12d 346	. . . . . . 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 2841	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝐵))
17		oveq2 7389	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (ω ↑o 𝑥) = (ω ↑o 𝐵))
18	17	oveq2d 7397	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝐴 ·o (ω ↑o 𝑥)) = (𝐴 ·o (ω ↑o 𝐵)))
19	18, 17	eqeq12d 2768	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥) ↔ (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵)))
20	16, 19	imbi12d 346	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)) ↔ (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
21		noel 4281	. . . . . . . . 9 ⊢ ¬ ∅ ∈ ∅
22	21	pm2.21i 119	. . . . . . . 8 ⊢ (∅ ∈ ∅ → (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅))
23	22	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ ∅ → (𝐴 ·o (ω ↑o ∅)) = (ω ↑o ∅)))
24		simprl 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ω ∈ On)
25		simpll 774	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ ω)
26		simplr 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ∅ ∈ 𝐴)
27		omabslem 8604	. . . . . . . . . . . . . . . 16 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω)
28	24, 25, 26, 27	syl3anc 1382	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o ω) = ω)
29	28	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·o ω) = ω)
30		suceq 6399	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ∅ → suc 𝑦 = suc ∅)
31		df-1o 8421	. . . . . . . . . . . . . . . . . 18 ⊢ 1o = suc ∅
32	30, 31	eqtr4di 2805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ∅ → suc 𝑦 = 1o)
33	32	oveq2d 7397	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ∅ → (ω ↑o suc 𝑦) = (ω ↑o 1o))
34		oe1 8497	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ On → (ω ↑o 1o) = ω)
35	34	ad2antrl 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑o 1o) = ω)
36	33, 35	sylan9eqr 2809	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (ω ↑o suc 𝑦) = ω)
37	36	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·o (ω ↑o suc 𝑦)) = (𝐴 ·o ω))
38	29, 37, 36	3eqtr4d 2797	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) ∧ 𝑦 = ∅) → (𝐴 ·o (ω ↑o suc 𝑦)) = (ω ↑o suc 𝑦))
39	38	ex 415	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ → (𝐴 ·o (ω ↑o suc 𝑦)) = (ω ↑o suc 𝑦)))
40	39	a1dd 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 7388	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) → ((𝐴 ·o (ω ↑o 𝑦)) ·o ω) = ((ω ↑o 𝑦) ·o ω))
42		oesuc 8480	. . . . . . . . . . . . . . . . . 18 ⊢ ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑o suc 𝑦) = ((ω ↑o 𝑦) ·o ω))
43	42	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑o suc 𝑦) = ((ω ↑o 𝑦) ·o ω))
44	43	oveq2d 7397	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o (ω ↑o suc 𝑦)) = (𝐴 ·o ((ω ↑o 𝑦) ·o ω)))
45		nnon 7837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
46	45	ad2antrr 734	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝐴 ∈ On)
47		oecl 8490	. . . . . . . . . . . . . . . . . 18 ⊢ ((ω ∈ On ∧ 𝑦 ∈ On) → (ω ↑o 𝑦) ∈ On)
48	47	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (ω ↑o 𝑦) ∈ On)
49		omass 8533	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ (ω ↑o 𝑦) ∈ On ∧ ω ∈ On) → ((𝐴 ·o (ω ↑o 𝑦)) ·o ω) = (𝐴 ·o ((ω ↑o 𝑦) ·o ω)))
50	46, 48, 24, 49	syl3anc 1382	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → ((𝐴 ·o (ω ↑o 𝑦)) ·o ω) = (𝐴 ·o ((ω ↑o 𝑦) ·o ω)))
51	44, 50	eqtr4d 2790	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝐴 ·o (ω ↑o suc 𝑦)) = ((𝐴 ·o (ω ↑o 𝑦)) ·o ω))
52	51, 43	eqeq12d 2768	. . . . . . . . . . . . . 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)))
53	41, 52	imbitrrid 248	. . . . . . . . . . . . 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)))
54	53	imim2d 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))))
55	54	com23 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 780	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → 𝑦 ∈ On)
57		on0eqel 6456	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝑦 ∈ On)) → (𝑦 = ∅ ∨ ∅ ∈ 𝑦))
59	40, 55, 58	mpjaod 869	. . . . . . . . . 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)))
60	59	a1dd 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))))
61	60	anassrs 470	. . . . . . . 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))))
62	61	expcom 416	. . . . . . 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)))))
63	45	ad3antrrr 738	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → 𝐴 ∈ On)
64		simprl 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ On)
65		simprr 780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → Lim 𝑥)
66		vex 3448	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑥 ∈ V
67	65, 66	jctil 526	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (𝑥 ∈ V ∧ Lim 𝑥))
68		limelon 6396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ∈ On)
69	67, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝑥 ∈ On)
70		oecl 8490	. . . . . . . . . . . . . . . 16 ⊢ ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ↑o 𝑥) ∈ On)
71	64, 69, 70	syl2anc 592	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (ω ↑o 𝑥) ∈ On)
72	71	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (ω ↑o 𝑥) ∈ On)
73		1onn 8594	. . . . . . . . . . . . . . . . 17 ⊢ 1o ∈ ω
74		ondif2 8455	. . . . . . . . . . . . . . . . 17 ⊢ (ω ∈ (On ∖ 2o) ↔ (ω ∈ On ∧ 1o ∈ ω))
75	64, 73, 74	sylanblrc 598	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ω ∈ (On ∖ 2o))
76	75	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ω ∈ (On ∖ 2o))
77	67	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝑥 ∈ V ∧ Lim 𝑥))
78		oelimcl 8554	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ (On ∖ 2o) ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → Lim (ω ↑o 𝑥))
79	76, 77, 78	syl2anc 592	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → Lim (ω ↑o 𝑥))
80		omlim 8486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ ((ω ↑o 𝑥) ∈ On ∧ Lim (ω ↑o 𝑥))) → (𝐴 ·o (ω ↑o 𝑥)) = ∪ 𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧))
81	63, 72, 79, 80	syl12anc 845	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝐴 ·o (ω ↑o 𝑥)) = ∪ 𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧))
82		simplrl 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ω ∈ On)
83		oelim2 8549	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ω ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (ω ↑o 𝑥) = ∪ 𝑦 ∈ (𝑥 ∖ 1o)(ω ↑o 𝑦))
84	82, 77, 83	syl2anc 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (ω ↑o 𝑥) = ∪ 𝑦 ∈ (𝑥 ∖ 1o)(ω ↑o 𝑦))
85	84	eleq2d 2838	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝑧 ∈ (ω ↑o 𝑥) ↔ 𝑧 ∈ ∪ 𝑦 ∈ (𝑥 ∖ 1o)(ω ↑o 𝑦)))
86		eliun 4943	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ (𝑥 ∖ 1o)(ω ↑o 𝑦) ↔ ∃𝑦 ∈ (𝑥 ∖ 1o)𝑧 ∈ (ω ↑o 𝑦))
87	85, 86	bitrdi 289	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝑧 ∈ (ω ↑o 𝑥) ↔ ∃𝑦 ∈ (𝑥 ∖ 1o)𝑧 ∈ (ω ↑o 𝑦)))
88	69	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → 𝑥 ∈ On)
89		anass 471	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦)) ↔ (𝑦 ∈ 𝑥 ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))))
90		onelon 6356	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On)
91		on0eln0 6388	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ On → (∅ ∈ 𝑦 ↔ 𝑦 ≠ ∅))
92	90, 91	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → (∅ ∈ 𝑦 ↔ 𝑦 ≠ ∅))
93	92	pm5.32da 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ≠ ∅)))
94		dif1o 8453	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝑥 ∖ 1o) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ≠ ∅))
95	93, 94	bitr4di 291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ↔ 𝑦 ∈ (𝑥 ∖ 1o)))
96	95	anbi1d 639	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → (((𝑦 ∈ 𝑥 ∧ ∅ ∈ 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦)) ↔ (𝑦 ∈ (𝑥 ∖ 1o) ∧ 𝑧 ∈ (ω ↑o 𝑦))))
97	89, 96	bitr3id 287	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ On → ((𝑦 ∈ 𝑥 ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))) ↔ (𝑦 ∈ (𝑥 ∖ 1o) ∧ 𝑧 ∈ (ω ↑o 𝑦))))
98	97	rexbidv2 3172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ On → (∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1o)𝑧 ∈ (ω ↑o 𝑦)))
99	88, 98	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦)) ↔ ∃𝑦 ∈ (𝑥 ∖ 1o)𝑧 ∈ (ω ↑o 𝑦)))
100	87, 99	bitr4d 284	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝑧 ∈ (ω ↑o 𝑥) ↔ ∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))))
101		r19.29 3115	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ ∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ∃𝑦 ∈ 𝑥 ((∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (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)))
103	102	imp 409	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ ∅ ∈ 𝑦) → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))
104	103	anim1i 623	. . . . . . . . . . . . . . . . . . . . 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)))
105	104	anasss 469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((∅ ∈ 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)))
106	71	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (ω ↑o 𝑥) ∈ On)
107		eloni 6341	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ω ↑o 𝑥) ∈ On → Ord (ω ↑o 𝑥))
108	106, 107	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → Ord (ω ↑o 𝑥))
109		simprr 780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑧 ∈ (ω ↑o 𝑦))
110	64	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ω ∈ On)
111	69	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑥 ∈ On)
112		simplr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑦 ∈ 𝑥)
113	111, 112, 90	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑦 ∈ On)
114	110, 113, 47	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (ω ↑o 𝑦) ∈ On)
115		onelon 6356	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((ω ↑o 𝑦) ∈ On ∧ 𝑧 ∈ (ω ↑o 𝑦)) → 𝑧 ∈ On)
116	114, 109, 115	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝑧 ∈ On)
117	45	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → 𝐴 ∈ On)
118	117	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → 𝐴 ∈ On)
119		simplr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ∅ ∈ 𝐴)
120	119	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ∅ ∈ 𝐴)
121		omord2 8520	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑧 ∈ On ∧ (ω ↑o 𝑦) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑧 ∈ (ω ↑o 𝑦) ↔ (𝐴 ·o 𝑧) ∈ (𝐴 ·o (ω ↑o 𝑦))))
122	116, 114, 118, 120, 121	syl31anc 1384	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝑧 ∈ (ω ↑o 𝑦) ↔ (𝐴 ·o 𝑧) ∈ (𝐴 ·o (ω ↑o 𝑦))))
123	109, 122	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ∈ (𝐴 ·o (ω ↑o 𝑦)))
124		simprl 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))
125	123, 124	eleqtrd 2854	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ∈ (ω ↑o 𝑦))
126	75	ad2antrr 734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → ω ∈ (On ∖ 2o))
127		oeord 8542	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ ω ∈ (On ∖ 2o)) → (𝑦 ∈ 𝑥 ↔ (ω ↑o 𝑦) ∈ (ω ↑o 𝑥)))
128	113, 111, 126, 127	syl3anc 1382	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝑦 ∈ 𝑥 ↔ (ω ↑o 𝑦) ∈ (ω ↑o 𝑥)))
129	112, 128	mpbid 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (ω ↑o 𝑦) ∈ (ω ↑o 𝑥))
130		ontr1 6378	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ω ↑o 𝑥) ∈ On → (((𝐴 ·o 𝑧) ∈ (ω ↑o 𝑦) ∧ (ω ↑o 𝑦) ∈ (ω ↑o 𝑥)) → (𝐴 ·o 𝑧) ∈ (ω ↑o 𝑥)))
131	106, 130	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((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)))
132	125, 129, 131	mp2and 707	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ∈ (ω ↑o 𝑥))
133		ordelss 6347	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord (ω ↑o 𝑥) ∧ (𝐴 ·o 𝑧) ∈ (ω ↑o 𝑥)) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
134	108, 132, 133	syl2anc 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) ∧ ((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
135	134	ex 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) → (((𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦) ∧ 𝑧 ∈ (ω ↑o 𝑦)) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥)))
136	105, 135	syl5 34	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ 𝑦 ∈ 𝑥) → (((∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥)))
137	136	rexlimdva 3153	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∃𝑦 ∈ 𝑥 ((∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥)))
138	101, 137	syl5 34	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → ((∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) ∧ ∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦))) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥)))
139	138	expdimp 455	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (∃𝑦 ∈ 𝑥 (∅ ∈ 𝑦 ∧ 𝑧 ∈ (ω ↑o 𝑦)) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥)))
140	100, 139	sylbid 242	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝑧 ∈ (ω ↑o 𝑥) → (𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥)))
141	140	ralrimiv 3143	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ∀𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
142		iunss 4992	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥) ↔ ∀𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
143	141, 142	sylibr 236	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ∪ 𝑧 ∈ (ω ↑o 𝑥)(𝐴 ·o 𝑧) ⊆ (ω ↑o 𝑥))
144	81, 143	eqsstrd 3961	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝐴 ·o (ω ↑o 𝑥)) ⊆ (ω ↑o 𝑥))
145		simpllr 783	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → ∅ ∈ 𝐴)
146		omword2 8527	. . . . . . . . . . . . 13 ⊢ ((((ω ↑o 𝑥) ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (ω ↑o 𝑥) ⊆ (𝐴 ·o (ω ↑o 𝑥)))
147	72, 63, 145, 146	syl21anc 846	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (ω ↑o 𝑥) ⊆ (𝐴 ·o (ω ↑o 𝑥)))
148	144, 147	eqssd 3944	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) ∧ ∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦))) → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥))
149	148	ex 415	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ Lim 𝑥)) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)))
150	149	anassrs 470	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)))
151	150	a1dd 50	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) ∧ Lim 𝑥) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥))))
152	151	expcom 416	. . . . . . 7 ⊢ (Lim 𝑥 → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∀𝑦 ∈ 𝑥 (∅ ∈ 𝑦 → (𝐴 ·o (ω ↑o 𝑦)) = (ω ↑o 𝑦)) → (∅ ∈ 𝑥 → (𝐴 ·o (ω ↑o 𝑥)) = (ω ↑o 𝑥)))))
153	5, 10, 15, 20, 23, 62, 152	tfinds3 7830	. . . . . 6 ⊢ (𝐵 ∈ On → (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
154	153	com12 32	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ ω ∈ On) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
155	154	adantrr 725	. . . 4 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐵 ∈ On → (∅ ∈ 𝐵 → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))))
156	155	imp32 421	. . 3 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))
157	156	an32s 660	. 2 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))
158		nnm0 8559	. . . 4 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
159	158	ad3antrrr 738	. . 3 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o ∅) = ∅)
160		fnoe 8463	. . . . . . 7 ⊢ ↑o Fn (On × On)
161		fndm 6609	. . . . . . 7 ⊢ ( ↑o Fn (On × On) → dom ↑o = (On × On))
162	160, 161	ax-mp 5	. . . . . 6 ⊢ dom ↑o = (On × On)
163	162	ndmov 7565	. . . . 5 ⊢ (¬ (ω ∈ On ∧ 𝐵 ∈ On) → (ω ↑o 𝐵) = ∅)
164	163	adantl 484	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (ω ↑o 𝐵) = ∅)
165	164	oveq2d 7397	. . 3 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (ω ↑o 𝐵)) = (𝐴 ·o ∅))
166	159, 165, 164	3eqtr4d 2797	. 2 ⊢ ((((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) ∧ ¬ (ω ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))
167	157, 166	pm2.61dan 820	1 ⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∀wral 3066  ∃wrex 3076  Vcvv 3444   ∖ cdif 3892   ⊆ wss 3895  ∅c0 4276  ∪ ciun 4939   × cxp 5634  dom cdm 5636  Ord word 6330  Oncon0 6331  Lim wlim 6332  suc csuc 6333   Fn wfn 6501  (class class class)co 7381  ωcom 7831  1_oc1o 8414  2_oc2o 8415   ·_o comu 8419   ↑_o coe 8420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-oadd 8425  df-omul 8426  df-oexp 8427

This theorem is referenced by:  cnfcom3  9645  omabs2  43847