Theorem unisalgen 45728

Description: The union of a set belongs to the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
unisalgen.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
unisalgen.s	⊢ 𝑆 = (SalGen‘𝑋)
unisalgen.u	⊢ 𝑈 = ∪ 𝑋

Assertion

Ref	Expression
unisalgen	⊢ (𝜑 → 𝑈 ∈ 𝑆)

Proof of Theorem unisalgen

Step	Hyp	Ref	Expression
1		unisalgen.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		unisalgen.s	. . . 4 ⊢ 𝑆 = (SalGen‘𝑋)
3		unisalgen.u	. . . 4 ⊢ 𝑈 = ∪ 𝑋
4	1, 2, 3	salgenuni 45725	. . 3 ⊢ (𝜑 → ∪ 𝑆 = 𝑈)
5	4	eqcomd 2734	. 2 ⊢ (𝜑 → 𝑈 = ∪ 𝑆)
6	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝑆 = (SalGen‘𝑋))
7		salgencl 45720	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) ∈ SAlg)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → (SalGen‘𝑋) ∈ SAlg)
9	6, 8	eqeltrd 2829	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
10		saluni 45713	. . 3 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)
11	9, 10	syl 17	. 2 ⊢ (𝜑 → ∪ 𝑆 ∈ 𝑆)
12	5, 11	eqeltrd 2829	1 ⊢ (𝜑 → 𝑈 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ∪ cuni 4908  ‘cfv 6548  SAlgcsalg 45696  SalGencsalgen 45700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-salg 45697  df-salgen 45701

This theorem is referenced by:  salgensscntex  45732