Theorem oeeulem 8227

Description: Lemma for oeeu 8229. (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 4103	. . . . . . . 8 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ∈ On)
3	2	adantl 484	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ On)
4		suceloni 7528	. . . . . . 7 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ∈ On)
6		oeworde 8219	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ suc 𝐵 ∈ On) → suc 𝐵 ⊆ (𝐴 ↑o suc 𝐵))
7	5, 6	syldan 593	. . . . . . 7 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝐵 ⊆ (𝐴 ↑o suc 𝐵))
8		sucidg 6269	. . . . . . . 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 3968	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ∈ (𝐴 ↑o suc 𝐵))
11		oveq2 7164	. . . . . . . 8 ⊢ (𝑥 = suc 𝐵 → (𝐴 ↑o 𝑥) = (𝐴 ↑o suc 𝐵))
12	11	eleq2d 2898	. . . . . . 7 ⊢ (𝑥 = suc 𝐵 → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o suc 𝐵)))
13	12	rspcev 3623	. . . . . 6 ⊢ ((suc 𝐵 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o suc 𝐵)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
14	5, 10, 13	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
15		onintrab2 7517	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥) ↔ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On)
16	14, 15	sylib 220	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On)
17		onuni 7508	. . . 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 2917	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ On)
20		sucidg 6269	. . . . . . 7 ⊢ (𝑋 ∈ On → 𝑋 ∈ suc 𝑋)
21	19, 20	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ suc 𝑋)
22		dif1o 8125	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ (On ∖ 1o) ↔ (𝐵 ∈ On ∧ 𝐵 ≠ ∅))
23	22	simprbi 499	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (On ∖ 1o) → 𝐵 ≠ ∅)
24	23	adantl 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐵 ≠ ∅)
25		ssrab2 4056	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ⊆ On
26		rabn0 4339	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ≠ ∅ ↔ ∃𝑥 ∈ On 𝐵 ∈ (𝐴 ↑o 𝑥))
27	14, 26	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ≠ ∅)
28		onint 7510	. . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
30		eleq1 2900	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ → ∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}))
32		oveq2 7164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → (𝐴 ↑o 𝑥) = (𝐴 ↑o ∅))
33	32	eleq2d 2898	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o ∅)))
34	33	elrab 3680	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (∅ ∈ On ∧ 𝐵 ∈ (𝐴 ↑o ∅)))
35	34	simprbi 499	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → 𝐵 ∈ (𝐴 ↑o ∅))
36		eldifi 4103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ (On ∖ 2o) → 𝐴 ∈ On)
37	36	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝐴 ∈ On)
38		oe0 8147	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (𝐴 ↑o ∅) = 1o)
39	37, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o ∅) = 1o)
40	39	eleq2d 2898	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴 ↑o ∅) ↔ 𝐵 ∈ 1o))
41		el1o 8124	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 1o ↔ 𝐵 = ∅)
42	40, 41	syl6bb 289	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐵 ∈ (𝐴 ↑o ∅) ↔ 𝐵 = ∅))
43	35, 42	syl5ib 246	. . . . . . . . . . . . 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 3029	. . . . . . . . . . 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 6251	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
48	47, 1	syl6eqr 2874	. . . . . . . . . . . . . . . 16 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = 𝑋)
49	48	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = 𝑋)
50	29	adantr 483	. . . . . . . . . . . . . . 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 2914	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
52		oveq2 7164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑋 → (𝐴 ↑o 𝑦) = (𝐴 ↑o 𝑋))
53	52	eleq2d 2898	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑋 → (𝐵 ∈ (𝐴 ↑o 𝑦) ↔ 𝐵 ∈ (𝐴 ↑o 𝑋)))
54		oveq2 7164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 ↑o 𝑥) = (𝐴 ↑o 𝑦))
55	54	eleq2d 2898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ (𝐴 ↑o 𝑥) ↔ 𝐵 ∈ (𝐴 ↑o 𝑦)))
56	55	cbvrabv 3491	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)}
57	53, 56	elrab2 3683	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o 𝑋)))
58	57	simprbi 499	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐴 ∈ On)
61		limeq 6203	. . . . . . . . . . . . . . . . 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 269	. . . . . . . . . . . . . . 15 ⊢ (Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} → Lim 𝑋)
64	19, 63	anim12i 614	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → (𝑋 ∈ On ∧ Lim 𝑋))
65		dif20el 8130	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (On ∖ 2o) → ∅ ∈ 𝐴)
66	65	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∅ ∈ 𝐴)
67		oelim 8159	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ (𝑋 ∈ On ∧ Lim 𝑋)) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝑋) = ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
68	60, 64, 66, 67	syl21anc 835	. . . . . . . . . . . . 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 2915	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦))
70		eliun 4923	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ∪ 𝑦 ∈ 𝑋 (𝐴 ↑o 𝑦) ↔ ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
71	69, 70	sylib 220	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → ∃𝑦 ∈ 𝑋 𝐵 ∈ (𝐴 ↑o 𝑦))
72	19	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) → 𝑋 ∈ On)
73		onss 7505	. . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) ∧ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ On)
76	49	eleq2d 2898	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 7519	. . . . . . . . . . . . 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 3270	. . . . . . . . . . 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 815	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
82		ioran 980	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∅ ∨ Lim ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)}))
84		eloni 6201	. . . . . . . . . 10 ⊢ (∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ∈ On → Ord ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
85		unizlim 6307	. . . . . . . . . 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 327	. . . . . . . 8 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ¬ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
88		orduniorsuc 7545	. . . . . . . . . 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 860	. . . . . . . 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 6256	. . . . . . . 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 2875	. . . . . 6 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 = ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
95	21, 94	eleqtrd 2915	. . . . 5 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
96	56	inteqi 4880	. . . . 5 ⊢ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} = ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)}
97	95, 96	eleqtrdi 2923	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → 𝑋 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑦)})
98	53	onnminsb 7519	. . . 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 8162	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
101	37, 19, 100	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ∈ On)
102		ontri1 6225	. . . 4 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ↑o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝑋)))
103	101, 3, 102	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ((𝐴 ↑o 𝑋) ⊆ 𝐵 ↔ ¬ 𝐵 ∈ (𝐴 ↑o 𝑋)))
104	99, 103	mpbird 259	. 2 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝐴 ↑o 𝑋) ⊆ 𝐵)
105	94, 29	eqeltrd 2913	. . 3 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)})
106		oveq2 7164	. . . . . 6 ⊢ (𝑦 = suc 𝑋 → (𝐴 ↑o 𝑦) = (𝐴 ↑o suc 𝑋))
107	106	eleq2d 2898	. . . . 5 ⊢ (𝑦 = suc 𝑋 → (𝐵 ∈ (𝐴 ↑o 𝑦) ↔ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
108	107, 56	elrab2 3683	. . . 4 ⊢ (suc 𝑋 ∈ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ↔ (suc 𝑋 ∈ On ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋)))
109	108	simprbi 499	. . 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 1124	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 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  {crab 3142   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  ∪ cuni 4838  ∩ cint 4876  ∪ ciun 4919  Ord word 6190  Oncon0 6191  Lim wlim 6192  suc csuc 6193  (class class class)co 7156  1_oc1o 8095  2_oc2o 8096   ↑_o coe 8101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-omul 8107  df-oexp 8108

This theorem is referenced by:  oeeui  8228  oeeu  8229