Theorem intsaluni 45855

Description: The union of an arbitrary intersection of sigma-algebras on the same set 𝑋, is 𝑋. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
intsaluni.ga	⊢ (𝜑 → 𝐺 ⊆ SAlg)
intsaluni.gn0	⊢ (𝜑 → 𝐺 ≠ ∅)
intsaluni.x	⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)

Assertion

Ref	Expression
intsaluni	⊢ (𝜑 → ∪ ∩ 𝐺 = 𝑋)

Distinct variable groups:   𝐺,𝑠   𝑋,𝑠   𝜑,𝑠

Proof of Theorem intsaluni

Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1909	. 2 ⊢ Ⅎ𝑠𝜑
2		nfv 1909	. 2 ⊢ Ⅎ𝑠∪ ∩ 𝐺 = 𝑋
3		intsaluni.gn0	. . 3 ⊢ (𝜑 → 𝐺 ≠ ∅)
4		n0 4346	. . . 4 ⊢ (𝐺 ≠ ∅ ↔ ∃𝑠 𝑠 ∈ 𝐺)
5	4	biimpi 215	. . 3 ⊢ (𝐺 ≠ ∅ → ∃𝑠 𝑠 ∈ 𝐺)
6	3, 5	syl 17	. 2 ⊢ (𝜑 → ∃𝑠 𝑠 ∈ 𝐺)
7		intss1 4967	. . . . . . 7 ⊢ (𝑠 ∈ 𝐺 → ∩ 𝐺 ⊆ 𝑠)
8	7	unissd 4919	. . . . . 6 ⊢ (𝑠 ∈ 𝐺 → ∪ ∩ 𝐺 ⊆ ∪ 𝑠)
9	8	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 ⊆ ∪ 𝑠)
10		intsaluni.x	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋)
11	9, 10	sseqtrd 4017	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 ⊆ 𝑋)
12	10	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 = 𝑋)
13		eleq1w 2808	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ 𝐺 ↔ 𝑡 ∈ 𝐺))
14	13	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → ((𝜑 ∧ 𝑠 ∈ 𝐺) ↔ (𝜑 ∧ 𝑡 ∈ 𝐺)))
15		unieq 4920	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑡 → ∪ 𝑠 = ∪ 𝑡)
16	15	eqeq1d 2727	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑡 → (∪ 𝑠 = 𝑋 ↔ ∪ 𝑡 = 𝑋))
17	14, 16	imbi12d 343	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 = 𝑋) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 = 𝑋)))
18	17, 10	chvarvv 1994	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 = 𝑋)
19	18	eqcomd 2731	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → 𝑋 = ∪ 𝑡)
20	19	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → 𝑋 = ∪ 𝑡)
21	12, 20	eqtrd 2765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 = ∪ 𝑡)
22		intsaluni.ga	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ SAlg)
23	22	sselda 3976	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → 𝑡 ∈ SAlg)
24		saluni 45851	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ SAlg → ∪ 𝑡 ∈ 𝑡)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 ∈ 𝑡)
26	25	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑡 ∈ 𝑡)
27	21, 26	eqeltrd 2825	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑡 ∈ 𝐺) → ∪ 𝑠 ∈ 𝑡)
28	27	ralrimiva 3135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡)
29		uniexg 7746	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝐺 → ∪ 𝑠 ∈ V)
30	29	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 ∈ V)
31		elintg 4958	. . . . . . . . 9 ⊢ (∪ 𝑠 ∈ V → (∪ 𝑠 ∈ ∩ 𝐺 ↔ ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡))
32	30, 31	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → (∪ 𝑠 ∈ ∩ 𝐺 ↔ ∀𝑡 ∈ 𝐺 ∪ 𝑠 ∈ 𝑡))
33	28, 32	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ 𝑠 ∈ ∩ 𝐺)
34	33	adantr 479	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → ∪ 𝑠 ∈ ∩ 𝐺)
35		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36	10	eqcomd 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → 𝑋 = ∪ 𝑠)
37	36	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑋 = ∪ 𝑠)
38	35, 37	eleqtrd 2827	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ 𝑠)
39		eleq2 2814	. . . . . . 7 ⊢ (𝑦 = ∪ 𝑠 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∪ 𝑠))
40	39	rspcev 3606	. . . . . 6 ⊢ ((∪ 𝑠 ∈ ∩ 𝐺 ∧ 𝑥 ∈ ∪ 𝑠) → ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
41	34, 38, 40	syl2anc 582	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
42		eluni2 4913	. . . . 5 ⊢ (𝑥 ∈ ∪ ∩ 𝐺 ↔ ∃𝑦 ∈ ∩ 𝐺𝑥 ∈ 𝑦)
43	41, 42	sylibr 233	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐺) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ∪ ∩ 𝐺)
44	11, 43	eqelssd 3998	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐺) → ∪ ∩ 𝐺 = 𝑋)
45	44	ex 411	. 2 ⊢ (𝜑 → (𝑠 ∈ 𝐺 → ∪ ∩ 𝐺 = 𝑋))
46	1, 2, 6, 45	exlimimdd 2207	1 ⊢ (𝜑 → ∪ ∩ 𝐺 = 𝑋)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  ∃wrex 3059  Vcvv 3461   ⊆ wss 3944  ∅c0 4322  ∪ cuni 4909  ∩ cint 4950  SAlgcsalg 45834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-ss 3961  df-nul 4323  df-pw 4606  df-uni 4910  df-int 4951  df-salg 45835

This theorem is referenced by:  intsal  45856  salgenuni  45863