insiga - Mathbox for Thierry Arnoux

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Theorem insiga 30200

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 4820	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	biimpi 206	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ∈ V)
3	2	adantr 481	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ V)
4		intssuni 4499	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
5	4	adantr 481	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ⊆ ∪ 𝐴)
6		simpr 477	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂))
7		elpwi 4168	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (sigAlgebra‘𝑂) → 𝐴 ⊆ (sigAlgebra‘𝑂))
8		sigasspw 30179	. . . . . . . 8 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑠 ⊆ 𝒫 𝑂)
9		selpw 4165	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑂 ↔ 𝑠 ⊆ 𝒫 𝑂)
10	8, 9	sylibr 224	. . . . . . 7 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑠 ∈ 𝒫 𝒫 𝑂)
11	10	ssriv 3607	. . . . . 6 ⊢ (sigAlgebra‘𝑂) ⊆ 𝒫 𝒫 𝑂
12	7, 11	syl6ss 3615	. . . . 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 4611	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝒫 𝑂 ↔ ∪ 𝐴 ⊆ 𝒫 𝑂)
15	13, 14	sylib 208	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∪ 𝐴 ⊆ 𝒫 𝑂)
16	5, 15	sstrd 3613	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ⊆ 𝒫 𝑂)
17		simpr 477	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
18		simplr 792	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂))
19		elelpwi 4171	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑠 ∈ (sigAlgebra‘𝑂))
20	17, 18, 19	syl2anc 693	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (sigAlgebra‘𝑂))
21		vex 3203	. . . . . . . . 9 ⊢ 𝑠 ∈ V
22		issiga 30174	. . . . . . . . 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 208	. . . . . . 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 479	. . . . . 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 1073	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑂 ∈ 𝑠)
27	26	ralrimiva 2966	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠)
28		n0 3931	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑠 𝑠 ∈ 𝐴)
29	28	biimpi 206	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ → ∃𝑠 𝑠 ∈ 𝐴)
30	29	adantr 481	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∃𝑠 𝑠 ∈ 𝐴)
31	20	ex 450	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑠 ∈ 𝐴 → 𝑠 ∈ (sigAlgebra‘𝑂)))
32	31	eximdv 1846	. . . . . . 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 6221	. . . . . . 7 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
35	34	exlimiv 1858	. . . . . 6 ⊢ (∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
36	33, 35	syl 17	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑂 ∈ V)
37		elintg 4483	. . . . 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 247	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑂 ∈ ∩ 𝐴)
40		simpll 790	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)))
41		simpr 477	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
42	40, 41	jca 554	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴))
43		elinti 4485	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝐴 → (𝑠 ∈ 𝐴 → 𝑥 ∈ 𝑠))
44	43	imp 445	. . . . . . . 8 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝑠)
45	44	adantll 750	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝑠)
46	25	simp2d 1074	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠)
47	46	r19.21bi 2932	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝑠) → (𝑂 ∖ 𝑥) ∈ 𝑠)
48	42, 45, 47	syl2anc 693	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝑂 ∖ 𝑥) ∈ 𝑠)
49	48	ralrimiva 2966	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠)
50		difexg 4808	. . . . . . . 8 ⊢ (𝑂 ∈ V → (𝑂 ∖ 𝑥) ∈ V)
51	36, 50	syl 17	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∖ 𝑥) ∈ V)
52	51	adantr 481	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑂 ∖ 𝑥) ∈ V)
53		elintg 4483	. . . . . 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 247	. . . 4 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑂 ∖ 𝑥) ∈ ∩ 𝐴)
56	55	ralrimiva 2966	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴)
57		simplll 798	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → (𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)))
58		simpr 477	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
59	57, 58	jca 554	. . . . . . . . 9 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴))
60		simpllr 799	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 ∩ 𝐴)
61		elpwi 4168	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ∩ 𝐴 → 𝑥 ⊆ ∩ 𝐴)
62		intss1 4492	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑠)
63	61, 62	sylan9ss 3616	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ⊆ 𝑠)
64		selpw 4165	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑠 ↔ 𝑥 ⊆ 𝑠)
65	63, 64	sylibr 224	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑠)
66	60, 65	sylancom 701	. . . . . . . . 9 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑠)
67	59, 66	jca 554	. . . . . . . 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 792	. . . . . . . 8 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ≼ ω)
69	25	simp3d 1075	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))
70	69	r19.21bi 2932	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))
71	67, 68, 70	sylc 65	. . . . . . 7 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → ∪ 𝑥 ∈ 𝑠)
72	71	ralrimiva 2966	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠)
73		uniexg 6955	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ∩ 𝐴 → ∪ 𝑥 ∈ V)
74	73	ad2antlr 763	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ V)
75		elintg 4483	. . . . . . 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 247	. . . . 5 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ ∩ 𝐴)
78	77	ex 450	. . . 4 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) → (𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴))
79	78	ralrimiva 2966	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴))
80	39, 56, 79	3jca 1242	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))
81		issiga 30174	. . 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 502	. 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 1324	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∪ cuni 4436 ∩ cint 4475 class class class wbr 4653 ‘cfv 5888 ωcom 7065 ≼ cdom 7953 sigAlgebracsiga 30170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-siga 30171

This theorem is referenced by: sigagensiga 30204

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