insiga - Mathbox for Thierry Arnoux

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Theorem insiga 34594

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 5316	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
2	1	birani 509	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ V)
3		intssuni 4937	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
4	3	adantr 486	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ⊆ ∪ 𝐴)
5		simpr 490	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂))
6		elpwi 4571	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 (sigAlgebra‘𝑂) → 𝐴 ⊆ (sigAlgebra‘𝑂))
7		sigasspw 34572	. . . . . . . 8 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑠 ⊆ 𝒫 𝑂)
8		velpw 4569	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝒫 𝑂 ↔ 𝑠 ⊆ 𝒫 𝑂)
9	7, 8	sylibr 237	. . . . . . 7 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑠 ∈ 𝒫 𝒫 𝑂)
10	9	ssriv 3942	. . . . . 6 ⊢ (sigAlgebra‘𝑂) ⊆ 𝒫 𝒫 𝑂
11	6, 10	sstrdi 3950	. . . . 5 ⊢ (𝐴 ∈ 𝒫 (sigAlgebra‘𝑂) → 𝐴 ⊆ 𝒫 𝒫 𝑂)
12	5, 11	syl 18	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝐴 ⊆ 𝒫 𝒫 𝑂)
13		sspwuni 5068	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝒫 𝑂 ↔ ∪ 𝐴 ⊆ 𝒫 𝑂)
14	12, 13	sylib 221	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∪ 𝐴 ⊆ 𝒫 𝑂)
15	4, 14	sstrd 3948	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ⊆ 𝒫 𝑂)
16		simpr 490	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
17		simplr 781	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂))
18		elelpwi 4574	. . . . . . . . 9 ⊢ ((𝑠 ∈ 𝐴 ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑠 ∈ (sigAlgebra‘𝑂))
19	16, 17, 18	syl2anc 596	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ (sigAlgebra‘𝑂))
20		vex 3461	. . . . . . . . 9 ⊢ 𝑠 ∈ V
21		issiga 34568	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∈ (sigAlgebra‘𝑂) ↔ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))))
22	20, 21	ax-mp 5	. . . . . . . 8 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) ↔ (𝑠 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))))
23	19, 22	sylib 221	. . . . . . 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))))
24	23	simprd 501	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → (𝑂 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠)))
25	24	simp1d 1160	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → 𝑂 ∈ 𝑠)
26	25	ralrimiva 3159	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠)
27		n0 4307	. . . . . . . 8 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑠 𝑠 ∈ 𝐴)
28	27	birani 509	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∃𝑠 𝑠 ∈ 𝐴)
29	19	ex 418	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑠 ∈ 𝐴 → 𝑠 ∈ (sigAlgebra‘𝑂)))
30	29	eximdv 1950	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (∃𝑠 𝑠 ∈ 𝐴 → ∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂)))
31	28, 30	mpd 16	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂))
32		elfvex 6920	. . . . . . 7 ⊢ (𝑠 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
33	32	exlimiv 1963	. . . . . 6 ⊢ (∃𝑠 𝑠 ∈ (sigAlgebra‘𝑂) → 𝑂 ∈ V)
34	31, 33	syl 18	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑂 ∈ V)
35		elintg 4922	. . . . 5 ⊢ (𝑂 ∈ V → (𝑂 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠))
36	34, 35	syl 18	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 𝑂 ∈ 𝑠))
37	26, 36	mpbird 260	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → 𝑂 ∈ ∩ 𝐴)
38		simpll 779	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)))
39		simpr 490	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
40	38, 39	jca 521	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴))
41		elinti 4923	. . . . . . . . 9 ⊢ (𝑥 ∈ ∩ 𝐴 → (𝑠 ∈ 𝐴 → 𝑥 ∈ 𝑠))
42	41	imp 412	. . . . . . . 8 ⊢ ((𝑥 ∈ ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝑠)
43	42	adantll 727	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝑠)
44	24	simp2d 1161	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠)
45	44	r19.21bi 3259	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝑠) → (𝑂 ∖ 𝑥) ∈ 𝑠)
46	40, 43, 45	syl2anc 596	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) ∧ 𝑠 ∈ 𝐴) → (𝑂 ∖ 𝑥) ∈ 𝑠)
47	46	ralrimiva 3159	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠)
48	34	difexd 5304	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∖ 𝑥) ∈ V)
49	48	adantr 486	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑂 ∖ 𝑥) ∈ V)
50		elintg 4922	. . . . . 6 ⊢ ((𝑂 ∖ 𝑥) ∈ V → ((𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠))
51	49, 50	syl 18	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → ((𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 (𝑂 ∖ 𝑥) ∈ 𝑠))
52	47, 51	mpbird 260	. . . 4 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ ∩ 𝐴) → (𝑂 ∖ 𝑥) ∈ ∩ 𝐴)
53	52	ralrimiva 3159	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴)
54		simplll 787	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → (𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)))
55		simpr 490	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑠 ∈ 𝐴)
56	54, 55	jca 521	. . . . . . . . 9 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴))
57		simpllr 788	. . . . . . . . . 10 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 ∩ 𝐴)
58		elpwi 4571	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 ∩ 𝐴 → 𝑥 ⊆ ∩ 𝐴)
59		intss1 4930	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → ∩ 𝐴 ⊆ 𝑠)
60	58, 59	sylan9ss 3951	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ⊆ 𝑠)
61		velpw 4569	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑠 ↔ 𝑥 ⊆ 𝑠)
62	60, 61	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 ∩ 𝐴 ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑠)
63	57, 62	sylancom 600	. . . . . . . . 9 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ∈ 𝒫 𝑠)
64	56, 63	jca 521	. . . . . . . 8 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝒫 𝑠))
65		simplr 781	. . . . . . . 8 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → 𝑥 ≼ ω)
66	24	simp3d 1162	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) → ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))
67	66	r19.21bi 3259	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑠 ∈ 𝐴) ∧ 𝑥 ∈ 𝒫 𝑠) → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑠))
68	64, 65, 67	sylc 66	. . . . . . 7 ⊢ (((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) ∧ 𝑠 ∈ 𝐴) → ∪ 𝑥 ∈ 𝑠)
69	68	ralrimiva 3159	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠)
70		uniexg 7748	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 ∩ 𝐴 → ∪ 𝑥 ∈ V)
71	70	ad2antlr 740	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ V)
72		elintg 4922	. . . . . . 7 ⊢ (∪ 𝑥 ∈ V → (∪ 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠))
73	71, 72	syl 18	. . . . . 6 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → (∪ 𝑥 ∈ ∩ 𝐴 ↔ ∀𝑠 ∈ 𝐴 ∪ 𝑥 ∈ 𝑠))
74	69, 73	mpbird 260	. . . . 5 ⊢ ((((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ ∩ 𝐴)
75	74	ex 418	. . . 4 ⊢ (((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) ∧ 𝑥 ∈ 𝒫 ∩ 𝐴) → (𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴))
76	75	ralrimiva 3159	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴))
77	37, 53, 76	3jca 1146	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))
78		issiga 34568	. . 3 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ (sigAlgebra‘𝑂) ↔ (∩ 𝐴 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))))
79	78	biimpar 483	. 2 ⊢ ((∩ 𝐴 ∈ V ∧ (∩ 𝐴 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ ∩ 𝐴(𝑂 ∖ 𝑥) ∈ ∩ 𝐴 ∧ ∀𝑥 ∈ 𝒫 ∩ 𝐴(𝑥 ≼ ω → ∪ 𝑥 ∈ ∩ 𝐴)))) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))
80	2, 15, 77, 79	syl12anc 850	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ∈ 𝒫 (sigAlgebra‘𝑂)) → ∩ 𝐴 ∈ (sigAlgebra‘𝑂))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∃wex 1812 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 Vcvv 3457 ∖ cdif 3903 ⊆ wss 3906 ∅c0 4286 𝒫 cpw 4564 ∪ cuni 4874 ∩ cint 4914 class class class wbr 5111 ‘cfv 6540 ωcom 7868 ≼ cdom 8947 sigAlgebracsiga 34564

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6496 df-fun 6542 df-fv 6548 df-siga 34565

This theorem is used by: sigagensiga 34598

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