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Theorem unisalgen 43554
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 (𝜑𝑋𝑉)
unisalgen.s 𝑆 = (SalGen‘𝑋)
unisalgen.u 𝑈 = 𝑋
Assertion
Ref Expression
unisalgen (𝜑𝑈𝑆)

Proof of Theorem unisalgen
StepHypRef Expression
1 unisalgen.x . . . 4 (𝜑𝑋𝑉)
2 unisalgen.s . . . 4 𝑆 = (SalGen‘𝑋)
3 unisalgen.u . . . 4 𝑈 = 𝑋
41, 2, 3salgenuni 43551 . . 3 (𝜑 𝑆 = 𝑈)
54eqcomd 2743 . 2 (𝜑𝑈 = 𝑆)
62a1i 11 . . . 4 (𝜑𝑆 = (SalGen‘𝑋))
7 salgencl 43546 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
81, 7syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
96, 8eqeltrd 2838 . . 3 (𝜑𝑆 ∈ SAlg)
10 saluni 43540 . . 3 (𝑆 ∈ SAlg → 𝑆𝑆)
119, 10syl 17 . 2 (𝜑 𝑆𝑆)
125, 11eqeltrd 2838 1 (𝜑𝑈𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110   cuni 4819  cfv 6380  SAlgcsalg 43524  SalGencsalgen 43528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-salg 43525  df-salgen 43529
This theorem is referenced by:  salgensscntex  43558
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