Theorem oeeulem 8637

Description: Lemma for oeeu 8639. (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 4140	. . . . . . . 8 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
4		onsuc 7830	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ∈ On)
6		oeworde 8629	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴 ↑o suc 𝐵))
7	5, 6	syldan 591	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ⊆ (𝐴 ↑o suc 𝐵))
8		sucidg 6466	. . . . . . . 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 3995	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴 ↑o suc 𝐵))
11		oveq2 7438	. . . . . . . 8 ⊢ (𝑥 = suc 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝐵))
12	11	eleq2d 2824	. . . . . . 7 ⊢ (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o suc 𝐵)))
13	12	rspcev 3621	. . . . . 6 ⊢ ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
14	5, 10, 13	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
15		onintrab2 7816	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On)
16	14, 15	sylib 218	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On)
17		onuni 7807	. . . 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	eqeltrid 2842	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
20		sucidg 6466	. . . . . . 7 ⊢ (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ suc 𝑋)
22		suceq 6451	. . . . . . . 8 ⊢ (𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → suc 𝑋 = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
23	1, 22	ax-mp 5	. . . . . . 7 ⊢ suc 𝑋 = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}
24		dif1o 8536	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (On ∖ 1o) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
25	24	simprbi 496	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ≠ ∅)
26	25	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ≠ ∅)
27		ssrab2 4089	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ⊆ On
28		rabn0 4394	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
29	14, 28	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ≠ ∅)
30		onint 7809	. . . . . . . . . . . . . . 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)})
31	27, 29, 30	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
32		eleq1 2826	. . . . . . . . . . . . . 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)}))
33	31, 32	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}))
34		oveq2 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
35	34	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o ∅)))
36	35	elrab 3694	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴 ↑o ∅)))
37	36	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → 𝐵 ∈ (𝐴 ↑o ∅))
38		eldifi 4140	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
39	38	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
40		oe0 8558	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
41	39, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o ∅) = 1o)
42	41	eleq2d 2824	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴 ↑o ∅) ↔ 𝐵 ∈ 1o))
43		el1o 8531	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
44	42, 43	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴 ↑o ∅) ↔ 𝐵 = ∅))
45	37, 44	imbitrid 244	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → 𝐵 = ∅))
46	33, 45	syld 47	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ → 𝐵 = ∅))
47	46	necon3ad 2950	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ≠ ∅ → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅))
48	26, 47	mpd 15	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅)
49		limuni 6446	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
50	49, 1	eqtr4di 2792	. . . . . . . . . . . . . . . 16 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = 𝑋)
51	50	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = 𝑋)
52	31	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)})
53	51, 52	eqeltrrd 2839	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
54		oveq2 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝐴 ↑o 𝑦) = (𝐴 ↑o 𝑋))
55	54	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝐵 ∈ (𝐴 ↑o 𝑦) ↔ 𝐵 ∈ (𝐴 ↑o 𝑋)))
56		oveq2 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
57	56	eleq2d 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o 𝑦)))
58	57	cbvrabv 3443	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)}
59	55, 58	elrab2 3697	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o 𝑋)))
60	59	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → 𝐵 ∈ (𝐴 ↑o 𝑋))
61	53, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐵 ∈ (𝐴 ↑o 𝑋))
62	38	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐴 ∈ On)
63		limeq 6397	. . . . . . . . . . . . . . . . 17 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = 𝑋 → (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ Lim 𝑋))
64	50, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ Lim 𝑋))
65	64	ibi 267	. . . . . . . . . . . . . . 15 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → Lim 𝑋)
66	19, 65	anim12i 613	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
67		dif20el 8541	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
68	67	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∅ ∈ 𝐴)
69		oelim 8570	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
70	62, 66, 68, 69	syl21anc 838	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → (𝐴 ↑o 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
71	61, 70	eleqtrd 2840	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
72		eliun 4999	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
73	71, 72	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
74	19	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ∈ On)
75		onss 7803	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ On → 𝑋 ⊆ On)
76	74, 75	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ⊆ On)
77	76	sselda 3994	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ On)
78	51	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ 𝑦 ∈ 𝑋))
79	78	biimpar 477	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
80	57	onnminsb 7818	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → (𝑦 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → ¬ 𝐵 ∈ (𝐴 ↑o 𝑦)))
81	77, 79, 80	sylc 65	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) ∧ 𝑦 ∈ 𝑋) → ¬ 𝐵 ∈ (𝐴 ↑o 𝑦))
82	81	nrexdv 3146	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ¬ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
83	73, 82	pm2.65da 817	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
84		ioran 985	. . . . . . . . . 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)}))
85	48, 83, 84	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}))
86		eloni 6395	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
87		unizlim 6508	. . . . . . . . . 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)})))
88	16, 86, 87	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)})))
89	85, 88	mtbird 325	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
90		orduniorsuc 7849	. . . . . . . . . 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)}))
91	16, 86, 90	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)}))
92	91	ord 864	. . . . . . . 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)}))
93	89, 92	mpd 15	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = suc ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
94	23, 93	eqtr4id 2793	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 = ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
95	21, 94	eleqtrd 2840	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
96	58	inteqi 4954	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)}
97	95, 96	eleqtrdi 2848	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)})
98	55	onnminsb 7818	. . . 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 8573	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
101	39, 19, 100	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ∈ On)
102		ontri1 6419	. . . 4 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝑋)))
103	101, 3, 102	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐴 ↑o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝑋)))
104	99, 103	mpbird 257	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ⊆ 𝐵)
105	94, 31	eqeltrd 2838	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
106		oveq2 7438	. . . . . 6 ⊢ (𝑦 = suc 𝑋 → (𝐴 ↑o 𝑦) = (𝐴 ↑o suc 𝑋))
107	106	eleq2d 2824	. . . . 5 ⊢ (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴 ↑o 𝑦) ↔ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
108	107, 58	elrab2 3697	. . . 4 ⊢ (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
109	108	simprbi 496	. . 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 1127	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∃wrex 3067  {crab 3432   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338  ∪ cuni 4911  ∩ cint 4950  ∪ ciun 4995  Ord word 6384  Oncon0 6385  Lim wlim 6386  suc csuc 6387  (class class class)co 7430  1_oc1o 8497  2_oc2o 8498   ↑_o coe 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-oexp 8510

This theorem is referenced by:  oeeui  8638  oeeu  8639