Theorem oeeulem 7568

Description: Lemma for oeeu 7570. (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 3694	. . . . . . . 8 ⊢ (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ∈ On)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ On)
4		suceloni 6905	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ∈ On)
6		oeworde 7560	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴 ↑𝑜 suc 𝐵))
7	5, 6	syldan 486	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝐵 ⊆ (𝐴 ↑𝑜 suc 𝐵))
8		sucidg 5720	. . . . . . . 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 3569	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐵))
11		oveq2 6557	. . . . . . . 8 ⊢ (𝑥 = suc 𝐵 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 suc 𝐵))
12	11	eleq2d 2673	. . . . . . 7 ⊢ (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐵)))
13	12	rspcev 3282	. . . . . 6 ⊢ ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥))
14	5, 10, 13	syl2anc 691	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥))
15		onintrab2 6894	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On)
16	14, 15	sylib 207	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On)
17		onuni 6885	. . . 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 2692	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ On)
20		sucidg 5720	. . . . . . 7 ⊢ (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ suc 𝑋)
22		dif1o 7467	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (On ∖ 1𝑜) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
23	22	simprbi 479	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (On ∖ 1𝑜) → 𝐵 ≠ ∅)
24	23	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐵 ≠ ∅)
25		ssrab2 3650	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ⊆ On
26		rabn0 3912	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑𝑜 𝑥))
27	14, 26	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ≠ ∅)
28		onint 6887	. . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
30		eleq1 2676	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
32		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 ∅))
33	32	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ↑𝑜 ∅)))
34	33	elrab 3331	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 ∅)))
35	34	simprbi 479	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → 𝐵 ∈ (𝐴 ↑𝑜 ∅))
36		eldifi 3694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → 𝐴 ∈ On)
37	36	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝐴 ∈ On)
38		oe0 7489	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (𝐴 ↑𝑜 ∅) = 1𝑜)
39	37, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 ∅) = 1𝑜)
40	39	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴 ↑𝑜 ∅) ↔ 𝐵 ∈ 1𝑜))
41		el1o 7466	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 1𝑜 ↔ 𝐵 = ∅)
42	40, 41	syl6bb 275	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐵 ∈ (𝐴 ↑𝑜 ∅) ↔ 𝐵 = ∅))
43	35, 42	syl5ib 233	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → 𝐵 = ∅))
44	31, 43	syld 46	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ → 𝐵 = ∅))
45	44	necon3ad 2795	. . . . . . . . . . 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 5702	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
48	47, 1	syl6eqr 2662	. . . . . . . . . . . . . . . 16 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = 𝑋)
49	48	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = 𝑋)
50	29	adantr 480	. . . . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
52		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝐴 ↑𝑜 𝑦) = (𝐴 ↑𝑜 𝑋))
53	52	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝐵 ∈ (𝐴 ↑𝑜 𝑦) ↔ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
54		oveq2 6557	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑜 𝑥) = (𝐴 ↑𝑜 𝑦))
55	54	eleq2d 2673	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ (𝐴 ↑𝑜 𝑥) ↔ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)))
56	55	cbvrabv 3172	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)}
57	53, 56	elrab2 3333	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
58	57	simprbi 479	. . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝐴 ∈ On)
61		limeq 5652	. . . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . 15 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → Lim 𝑋)
64	19, 63	anim12i 588	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
65		dif20el 7472	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (On ∖ 2𝑜) → ∅ ∈ 𝐴)
66	65	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∅ ∈ 𝐴)
67		oelim 7501	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦))
68	60, 64, 66, 67	syl21anc 1317	. . . . . . . . . . . . 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 2690	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦))
70		eliun 4460	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑𝑜 𝑦) ↔ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
71	69, 70	sylib 207	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
72	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) → 𝑋 ∈ On)
73		onss 6882	. . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ On)
76	49	eleq2d 2673	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)))
79	75, 77, 78	sylc 63	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}) ∧ 𝑦 ∈ 𝑋) → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑦))
80	79	nrexdv 2984	. . . . . . . . . . 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 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
82		ioran 510	. . . . . . . . . 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 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)}))
84		eloni 5650	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
85		unizlim 5761	. . . . . . . . . 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 314	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
88		orduniorsuc 6922	. . . . . . . . . 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 391	. . . . . . . 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 5707	. . . . . . . 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 2663	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 = ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
95	21, 94	eleqtrd 2690	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
96	56	inteqi 4414	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} = ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)}
97	95, 96	syl6eleq 2698	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → 𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)})
98	53	onnminsb 6896	. . . 4 ⊢ (𝑋 ∈ On → (𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑦)} → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
99	19, 97, 98	sylc 63	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋))
100		oecl 7504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑𝑜 𝑋) ∈ On)
101	37, 19, 100	syl2anc 691	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ∈ On)
102		ontri1 5674	. . . 4 ⊢ (((𝐴 ↑𝑜 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
103	101, 3, 102	syl2anc 691	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ((𝐴 ↑𝑜 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑𝑜 𝑋)))
104	99, 103	mpbird 246	. 2 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (𝐴 ↑𝑜 𝑋) ⊆ 𝐵)
105	94, 29	eqeltrd 2688	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)})
106		oveq2 6557	. . . . . 6 ⊢ (𝑦 = suc 𝑋 → (𝐴 ↑𝑜 𝑦) = (𝐴 ↑𝑜 suc 𝑋))
107	106	eleq2d 2673	. . . . 5 ⊢ (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴 ↑𝑜 𝑦) ↔ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))
108	107, 56	elrab2 3333	. . . 4 ⊢ (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑𝑜 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑𝑜 suc 𝑋)))
109	108	simprbi 479	. . 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 1235	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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ∩ cint 4410  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oeeui  7569  oeeu  7570