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Theorem unisalgen 44829
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 44826 . . 3 (𝜑 𝑆 = 𝑈)
54eqcomd 2737 . 2 (𝜑𝑈 = 𝑆)
62a1i 11 . . . 4 (𝜑𝑆 = (SalGen‘𝑋))
7 salgencl 44821 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
81, 7syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
96, 8eqeltrd 2832 . . 3 (𝜑𝑆 ∈ SAlg)
10 saluni 44814 . . 3 (𝑆 ∈ SAlg → 𝑆𝑆)
119, 10syl 17 . 2 (𝜑 𝑆𝑆)
125, 11eqeltrd 2832 1 (𝜑𝑈𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106   cuni 4901  cfv 6532  SAlgcsalg 44797  SalGencsalgen 44801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-salg 44798  df-salgen 44802
This theorem is referenced by:  salgensscntex  44833
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