Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unisalgen Structured version   Visualization version   GIF version

Theorem unisalgen 45046
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 45043 . . 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 45038 . . . . 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 45031 . . 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 1541   ∈ wcel 2106  βˆͺ cuni 4908  β€˜cfv 6543  SAlgcsalg 45014  SalGencsalgen 45018
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 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 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-salg 45015  df-salgen 45019
This theorem is referenced by:  salgensscntex  45050
  Copyright terms: Public domain W3C validator