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Theorem unisalgen 39234
 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 39231 . . 3 (𝜑 𝑆 = 𝑈)
54eqcomd 2616 . 2 (𝜑𝑈 = 𝑆)
62a1i 11 . . . 4 (𝜑𝑆 = (SalGen‘𝑋))
7 salgencl 39226 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
81, 7syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
96, 8eqeltrd 2688 . . 3 (𝜑𝑆 ∈ SAlg)
10 saluni 39220 . . 3 (𝑆 ∈ SAlg → 𝑆𝑆)
119, 10syl 17 . 2 (𝜑 𝑆𝑆)
125, 11eqeltrd 2688 1 (𝜑𝑈𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372  ‘cfv 5804  SAlgcsalg 39204  SalGencsalgen 39208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-salg 39205  df-salgen 39209 This theorem is referenced by:  salgensscntex  39238
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