insiga - Mathbox for Thierry Arnoux

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Theorem insiga 29527

Description: The intersection of a collection of sigma-algebras of same base is a sigma-algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)

**Assertion**
Ref	Expression
insiga	⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))

Proof of Theorem insiga Dummy variables 𝑥 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		intex 4747	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	biimpi 205	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ V)
4		intssuni 4434	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantr 480	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ⊆ ∪ 𝐴)
6		simpr 476	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂))
7		elpwi 4117	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (sigAlgebra‘𝑂) → 𝐴 ⊆ (sigAlgebra‘𝑂))
8		sigasspw 29506	. . . . . . . 8 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑠 ⊆ 𝒫 𝑂)
9		selpw 4115	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑂 ↔ 𝑠 ⊆ 𝒫 𝑂)
10	8, 9	sylibr 223	. . . . . . 7 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑠 ∈ 𝒫 𝒫 𝑂)
11	10	ssriv 3572	. . . . . 6 ⊢ (sigAlgebra‘𝑂) ⊆ 𝒫 𝒫 𝑂
12	7, 11	syl6ss 3580	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (sigAlgebra‘𝑂) → 𝐴 ⊆ 𝒫 𝒫 𝑂)
13	6, 12	syl 17	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝐴 ⊆ 𝒫 𝒫 𝑂)
14		sspwuni 4547	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝒫 𝑂 ↔ ∪ 𝐴 ⊆ 𝒫 𝑂)
15	13, 14	sylib 207	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∪ 𝐴 ⊆ 𝒫 𝑂)
16	5, 15	sstrd 3578	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ⊆ 𝒫 𝑂)
17		simpr 476	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
18		simplr 788	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂))
19		elelpwi 4119	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑠 ∈ (sigAlgebra‘𝑂))
20	17, 18, 19	syl2anc 691	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (sigAlgebra‘𝑂))
21		vex 3176	. . . . . . . . 9 ⊢ 𝑠 ∈ V
22		issiga 29501	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ (sigAlgebra‘𝑂) ↔ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) ↔ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))))
24	20, 23	sylib 207	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))))
25	24	simprd 478	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))
26	25	simp1d 1066	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑂 ∈ 𝑠)
27	26	ralrimiva 2949	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠)
28		n0 3890	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑠 𝑠 ∈ 𝐴)
29	28	biimpi 205	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → ∃𝑠 𝑠 ∈ 𝐴)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∃𝑠 𝑠 ∈ 𝐴)
31	20	ex 449	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑠 ∈ 𝐴 → 𝑠 ∈ (sigAlgebra‘𝑂)))
32	31	eximdv 1833	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (∃𝑠 𝑠 ∈ 𝐴 → ∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂)))
33	30, 32	mpd 15	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂))
34		elfvex 6131	. . . . . . 7 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
35	34	exlimiv 1845	. . . . . 6 ⊢ (∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
36	33, 35	syl 17	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑂 ∈ V)
37		elintg 4418	. . . . 5 ⊢ (𝑂 ∈ V → (𝑂 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠))
38	36, 37	syl 17	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠))
39	27, 38	mpbird 246	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑂 ∈ ∩ 𝐴)
40		simpll 786	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)))
41		simpr 476	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
42	40, 41	jca 553	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴))
43		elinti 4420	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝐴 → (𝑠 ∈ 𝐴 → 𝑥 ∈ 𝑠))
44	43	imp 444	. . . . . . . 8 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝑠)
45	44	adantll 746	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝑠)
46	25	simp2d 1067	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠)
47	46	r19.21bi 2916	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝑠) → (𝑂 ∖ 𝑥) ∈ 𝑠)
48	42, 45, 47	syl2anc 691	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝑂 ∖ 𝑥) ∈ 𝑠)
49	48	ralrimiva 2949	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠)
50		difexg 4735	. . . . . . . 8 ⊢ (𝑂 ∈ V → (𝑂 ∖ 𝑥) ∈ V)
51	36, 50	syl 17	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∖ 𝑥) ∈ V)
52	51	adantr 480	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑂 ∖ 𝑥) ∈ V)
53		elintg 4418	. . . . . 6 ⊢ ((𝑂 ∖ 𝑥) ∈ V → ((𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠))
54	52, 53	syl 17	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → ((𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠))
55	49, 54	mpbird 246	. . . 4 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑂 ∖ 𝑥) ∈ ∩ 𝐴)
56	55	ralrimiva 2949	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴)
57		simplll 794	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → (𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)))
58		simpr 476	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
59	57, 58	jca 553	. . . . . . . . 9 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴))
60		simpllr 795	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 ∩ 𝐴)
61		elpwi 4117	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ∩ 𝐴 → 𝑥 ⊆ ∩ 𝐴)
62		intss1 4427	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑠)
63	61, 62	sylan9ss 3581	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ⊆ 𝑠)
64		selpw 4115	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑠 ↔ 𝑥 ⊆ 𝑠)
65	63, 64	sylibr 223	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑠)
66	60, 65	sylancom 698	. . . . . . . . 9 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑠)
67	59, 66	jca 553	. . . . . . . 8 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝒫 𝑠))
68		simplr 788	. . . . . . . 8 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ≼ ω)
69	25	simp3d 1068	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))
70	69	r19.21bi 2916	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))
71	67, 68, 70	sylc 63	. . . . . . 7 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → ∪ 𝑥 ∈ 𝑠)
72	71	ralrimiva 2949	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠)
73		uniexg 6853	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ∩ 𝐴 → ∪ 𝑥 ∈ V)
74	73	ad2antlr 759	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ V)
75		elintg 4418	. . . . . . 7 ⊢ (∪ 𝑥 ∈ V → (∪ 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠))
76	74, 75	syl 17	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → (∪ 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠))
77	72, 76	mpbird 246	. . . . 5 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ ∩ 𝐴)
78	77	ex 449	. . . 4 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) → (𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴))
79	78	ralrimiva 2949	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴))
80	39, 56, 79	3jca 1235	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))
81		issiga 29501	. . 3 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ (sigAlgebra‘𝑂) ↔ (∩ 𝐴 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))))
82	81	biimpar 501	. 2 ⊢ ((∩ 𝐴 ∈ V ∧ (∩ 𝐴 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))
83	3, 16, 80, 82	syl12anc 1316	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 ∩ cint 4410 class class class wbr 4583 ‘cfv 5804 ωcom 6957 ≼ cdom 7839 sigAlgebracsiga 29497

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-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-3an 1033 df-tru 1478 df-fal 1481 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-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-siga 29498

This theorem is referenced by: sigagensiga 29531

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