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Theorem unisalgen 44671
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 44668 . . 3 (πœ‘ β†’ βˆͺ 𝑆 = π‘ˆ)
54eqcomd 2738 . 2 (πœ‘ β†’ π‘ˆ = βˆͺ 𝑆)
62a1i 11 . . . 4 (πœ‘ β†’ 𝑆 = (SalGenβ€˜π‘‹))
7 salgencl 44663 . . . . 5 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
81, 7syl 17 . . . 4 (πœ‘ β†’ (SalGenβ€˜π‘‹) ∈ SAlg)
96, 8eqeltrd 2833 . . 3 (πœ‘ β†’ 𝑆 ∈ SAlg)
10 saluni 44656 . . 3 (𝑆 ∈ SAlg β†’ βˆͺ 𝑆 ∈ 𝑆)
119, 10syl 17 . 2 (πœ‘ β†’ βˆͺ 𝑆 ∈ 𝑆)
125, 11eqeltrd 2833 1 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4869  β€˜cfv 6500  SAlgcsalg 44639  SalGencsalgen 44643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2703  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-salg 44640  df-salgen 44644
This theorem is referenced by:  salgensscntex  44675
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