Theorem oeeulem 7681

Description: Lemma for oeeu 7683. (Contributed by Mario Carneiro, 28-Feb-2013.)

Hypothesis

Ref	Expression
oeeu.1	⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}

Assertion

Ref	Expression
oeeulem	⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem oeeulem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oeeu.1	. . 3 ⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}
2		eldifi 3732	. . . . . . . 8 ⊢ (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ∈ On)
3	2	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ On)
4		suceloni 7013	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ∈ On)
6		oeworde 7673	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴 ↑𝑜 suc 𝐵))
7	5, 6	syldan 487	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ⊆ (𝐴 ↑𝑜 suc 𝐵))
8		sucidg 5803	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
9	3, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ suc 𝐵)
10	7, 9	sseldd 3604	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐵))
11		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = suc 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝐵))
12	11	eleq2d 2687	. . . . . . 7 ⊢ (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐵)))
13	12	rspcev 3309	. . . . . 6 ⊢ ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥))
14	5, 10, 13	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥))
15		onintrab2 7002	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On)
16	14, 15	sylib 208	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On)
17		onuni 6993	. . . 4 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On → ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On)
18	16, 17	syl 17	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On)
19	1, 18	syl5eqel 2705	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ On)
20		sucidg 5803	. . . . . . 7 ⊢ (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ suc 𝑋)
22		dif1o 7580	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (On ∖ 1𝑜) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
23	22	simprbi 480	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ≠ ∅)
24	23	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ≠ ∅)
25		ssrab2 3687	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ⊆ On
26		rabn0 3958	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥))
27	14, 26	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ≠ ∅)
28		onint 6995	. . . . . . . . . . . . . . 15 ⊢ (({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ⊆ On ∧ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ≠ ∅) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
29	25, 27, 28	sylancr 695	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
30		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
31	29, 30	syl5ibcom 235	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
32		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
33	32	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ↑𝑜 ∅)))
34	33	elrab 3363	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 ∅)))
35	34	simprbi 480	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → 𝐵 ∈ (𝐴 ↑𝑜 ∅))
36		eldifi 3732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
37	36	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐴 ∈ On)
38		oe0 7602	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
39	37, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 ∅) = 1𝑜)
40	39	eleq2d 2687	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴 ↑𝑜 ∅) ↔ 𝐵 ∈ 1𝑜))
41		el1o 7579	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 1𝑜 ↔ 𝐵 = ∅)
42	40, 41	syl6bb 276	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴 ↑𝑜 ∅) ↔ 𝐵 = ∅))
43	35, 42	syl5ib 234	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → 𝐵 = ∅))
44	31, 43	syld 47	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ → 𝐵 = ∅))
45	44	necon3ad 2807	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ≠ ∅ → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅))
46	24, 45	mpd 15	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅)
47		limuni 5785	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
48	47, 1	syl6eqr 2674	. . . . . . . . . . . . . . . 16 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = 𝑋)
49	48	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = 𝑋)
50	29	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
51	49, 50	eqeltrrd 2702	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
52		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝐴 ↑𝑜 𝑦) = (𝐴 ↑𝑜 𝑋))
53	52	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝐵 ∈ (𝐴 ↑𝑜 𝑦) ↔ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
54		oveq2 6658	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
55	54	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)))
56	55	cbvrabv 3199	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)}
57	53, 56	elrab2 3366	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
58	57	simprbi 480	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → 𝐵 ∈ (𝐴 ↑𝑜 𝑋))
59	51, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝐵 ∈ (𝐴 ↑𝑜 𝑋))
60	36	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝐴 ∈ On)
61		limeq 5735	. . . . . . . . . . . . . . . . 17 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = 𝑋 → (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ Lim 𝑋))
62	48, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ Lim 𝑋))
63	62	ibi 256	. . . . . . . . . . . . . . 15 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → Lim 𝑋)
64	19, 63	anim12i 590	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
65		dif20el 7585	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
66	65	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∅ ∈ 𝐴)
67		oelim 7614	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦))
68	60, 64, 66, 67	syl21anc 1325	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → (𝐴 ↑𝑜 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦))
69	59, 68	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦))
70		eliun 4524	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦) ↔ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
71	69, 70	sylib 208	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
72	19	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝑋 ∈ On)
73		onss 6990	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ On → 𝑋 ⊆ On)
74	72, 73	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝑋 ⊆ On)
75	74	sselda 3603	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ On)
76	49	eleq2d 2687	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ 𝑦 ∈ 𝑋))
77	76	biimpar 502	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
78	55	onnminsb 7004	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)))
79	75, 77, 78	sylc 65	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) ∧ 𝑦 ∈ 𝑋) → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
80	79	nrexdv 3001	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ¬ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
81	71, 80	pm2.65da 600	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
82		ioran 511	. . . . . . . . . 10 ⊢ (¬ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) ↔ (¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ ∧ ¬ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
83	46, 81, 82	sylanbrc 698	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
84		eloni 5733	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
85		unizlim 5844	. . . . . . . . . 10 ⊢ (Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})))
86	16, 84, 85	3syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})))
87	83, 86	mtbird 315	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
88		orduniorsuc 7030	. . . . . . . . . 10 ⊢ (Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∨ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
89	16, 84, 88	3syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∨ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
90	89	ord 392	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
91	87, 90	mpd 15	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
92		suceq 5790	. . . . . . . 8 ⊢ (𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → suc 𝑋 = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
93	1, 92	ax-mp 5	. . . . . . 7 ⊢ suc 𝑋 = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}
94	91, 93	syl6reqr 2675	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 = ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
95	21, 94	eleqtrd 2703	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
96	56	inteqi 4479	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)}
97	95, 96	syl6eleq 2711	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)})
98	53	onnminsb 7004	. . . 4 ⊢ (𝑋 ∈ On → (𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)} → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
99	19, 97, 98	sylc 65	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋))
100		oecl 7617	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑𝑜 𝑋) ∈ On)
101	37, 19, 100	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ∈ On)
102		ontri1 5757	. . . 4 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
103	101, 3, 102	syl2anc 693	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
104	99, 103	mpbird 247	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ⊆ 𝐵)
105	94, 29	eqeltrd 2701	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
106		oveq2 6658	. . . . . 6 ⊢ (𝑦 = suc 𝑋 → (𝐴 ↑𝑜 𝑦) = (𝐴 ↑𝑜 suc 𝑋))
107	106	eleq2d 2687	. . . . 5 ⊢ (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴 ↑𝑜 𝑦) ↔ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))
108	107, 56	elrab2 3366	. . . 4 ⊢ (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))
109	108	simprbi 480	. . 3 ⊢ (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋))
110	105, 109	syl 17	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋))
111	19, 104, 110	3jca 1242	1 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝑋 ∈ On ∧ (𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ∩ cint 4475  ∪ ciun 4520  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑜 coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oeeui  7682  oeeu  7683