Theorem oeeulem 8527

Description: Lemma for oeeu 8529. (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 4061	. . . . . . . 8 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
3	2	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
4		onsuc 7753	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ∈ On)
6		oeworde 8519	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴 ↑o suc 𝐵))
7	5, 6	syldan 597	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ⊆ (𝐴 ↑o suc 𝐵))
8		sucidg 6393	. . . . . . . 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 3916	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴 ↑o suc 𝐵))
11		oveq2 7364	. . . . . . . 8 ⊢ (𝑥 = suc 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝐵))
12	11	eleq2d 2825	. . . . . . 7 ⊢ (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o suc 𝐵)))
13	12	rspcev 3560	. . . . . 6 ⊢ ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
14	5, 10, 13	syl2anc 590	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
15		onintrab2 7740	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On)
16	14, 15	sylib 219	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On)
17		onuni 7731	. . . 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 2843	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
20		sucidg 6393	. . . . . . 7 ⊢ (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ suc 𝑋)
22		suceq 6378	. . . . . . . 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 8425	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (On ∖ 1o) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
25	24	simprbi 498	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ≠ ∅)
26	25	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ≠ ∅)
27		ssrab2 4011	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ⊆ On
28		rabn0 4317	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
29	14, 28	sylibr 235	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ≠ ∅)
30		onint 7733	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
32		eleq1 2827	. . . . . . . . . . . . . 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 246	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}))
34		oveq2 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
35	34	eleq2d 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o ∅)))
36	35	elrab 3629	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴 ↑o ∅)))
37	36	simprbi 498	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → 𝐵 ∈ (𝐴 ↑o ∅))
38		eldifi 4061	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
39	38	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
40		oe0 8447	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
41	39, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o ∅) = 1o)
42	41	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴 ↑o ∅) ↔ 𝐵 ∈ 1o))
43		el1o 8420	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
44	42, 43	bitrdi 288	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴 ↑o ∅) ↔ 𝐵 = ∅))
45	37, 44	imbitrid 245	. . . . . . . . . . . . 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 2947	. . . . . . . . . . 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 6372	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = 𝑋)
52	31	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)})
53	51, 52	eqeltrrd 2840	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
54		oveq2 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝐴 ↑o 𝑦) = (𝐴 ↑o 𝑋))
55	54	eleq2d 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝐵 ∈ (𝐴 ↑o 𝑦) ↔ 𝐵 ∈ (𝐴 ↑o 𝑋)))
56		oveq2 7364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
57	56	eleq2d 2825	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o 𝑦)))
58	57	cbvrabv 3401	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)}
59	55, 58	elrab2 3632	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o 𝑋)))
60	59	simprbi 498	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐴 ∈ On)
63		limeq 6322	. . . . . . . . . . . . . . . . 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 268	. . . . . . . . . . . . . . 15 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → Lim 𝑋)
66	19, 65	anim12i 619	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
67		dif20el 8430	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
68	67	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∅ ∈ 𝐴)
69		oelim 8459	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
70	62, 66, 68, 69	syl21anc 843	. . . . . . . . . . . . 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 2841	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
72		eliun 4925	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
73	71, 72	sylib 219	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
74	19	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ∈ On)
75		onss 7728	. . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ On)
78	51	eleq2d 2825	. . . . . . . . . . . . . 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 478	. . . . . . . . . . . . 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 7742	. . . . . . . . . . . . 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 3134	. . . . . . . . . . 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 822	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
84		ioran 991	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}))
86		eloni 6320	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
87		unizlim 6434	. . . . . . . . . 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 326	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
90		orduniorsuc 7770	. . . . . . . . . 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 870	. . . . . . . 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 2841	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
96	58	inteqi 4881	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)}
97	95, 96	eleqtrdi 2849	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)})
98	55	onnminsb 7742	. . . 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 8462	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
101	39, 19, 100	syl2anc 590	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ∈ On)
102		ontri1 6344	. . . 4 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝑋)))
103	101, 3, 102	syl2anc 590	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐴 ↑o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝑋)))
104	99, 103	mpbird 258	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ⊆ 𝐵)
105	94, 31	eqeltrd 2839	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
106		oveq2 7364	. . . . . 6 ⊢ (𝑦 = suc 𝑋 → (𝐴 ↑o 𝑦) = (𝐴 ↑o suc 𝑋))
107	106	eleq2d 2825	. . . . 5 ⊢ (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴 ↑o 𝑦) ↔ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
108	107, 58	elrab2 3632	. . . 4 ⊢ (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
109	108	simprbi 498	. . 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 1134	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 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4261  ∪ cuni 4838  ∩ cint 4877  ∪ ciun 4921  Ord word 6309  Oncon0 6310  Lim wlim 6311  suc csuc 6312  (class class class)co 7356  1_oc1o 8388  2_oc2o 8389   ↑_o coe 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-omul 8400  df-oexp 8401

This theorem is referenced by:  oeeui  8528  oeeu  8529