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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 (πœ‘ β†’ 𝑋 ∈ 𝑉)
unisalgen.s 𝑆 = (SalGenβ€˜π‘‹)
unisalgen.u π‘ˆ = βˆͺ 𝑋
Assertion
Ref Expression
unisalgen (πœ‘ β†’ π‘ˆ ∈ 𝑆)

Proof of Theorem unisalgen
StepHypRef Expression
1 unisalgen.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2 unisalgen.s . . . 4 𝑆 = (SalGenβ€˜π‘‹)
3 unisalgen.u . . . 4 π‘ˆ = βˆͺ 𝑋
41, 2, 3salgenuni 45725 . . 3 (πœ‘ β†’ βˆͺ 𝑆 = π‘ˆ)
54eqcomd 2734 . 2 (πœ‘ β†’ π‘ˆ = βˆͺ 𝑆)
62a1i 11 . . . 4 (πœ‘ β†’ 𝑆 = (SalGenβ€˜π‘‹))
7 salgencl 45720 . . . . 5 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
81, 7syl 17 . . . 4 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
96, 8eqeltrd 2829 . . 3 (πœ‘ β†’ 𝑆 ∈ SAlg)
10 saluni 45713 . . 3 (𝑆 ∈ SAlg β†’ βˆͺ 𝑆 ∈ 𝑆)
119, 10syl 17 . 2 (πœ‘ β†’ βˆͺ 𝑆 ∈ 𝑆)
125, 11eqeltrd 2829 1 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆͺ cuni 4908  β€˜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
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