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Theorem elsigagen2 24523
 Description: Any countable union of elements of a set is also in the sigma algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
Assertion
Ref Expression
elsigagen2 sigaGen

Proof of Theorem elsigagen2
StepHypRef Expression
1 simp1 957 . . 3
21sgsiga 24517 . 2 sigaGen sigAlgebra
3 sssigagen 24520 . . . 4 sigaGen
4 sspwb 4405 . . . . 5 sigaGen sigaGen
54biimpi 187 . . . 4 sigaGen sigaGen
61, 3, 53syl 19 . . 3 sigaGen
7 simp2 958 . . . 4
8 simp3 959 . . . . 5
9 ctex 24092 . . . . 5
10 elpwg 3798 . . . . 5
118, 9, 103syl 19 . . . 4
127, 11mpbird 224 . . 3
136, 12sseldd 3341 . 2 sigaGen
14 sigaclcu 24492 . 2 sigaGen sigAlgebra sigaGen sigaGen
152, 13, 8, 14syl3anc 1184 1 sigaGen
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   w3a 936   wcel 1725  cvv 2948   wss 3312  cpw 3791  cuni 4007   class class class wbr 4204  com 4837   crn 4871  cfv 5446   cdom 7099  sigAlgebracsiga 24482  sigaGencsigagen 24513 This theorem is referenced by:  sxbrsigalem1  24627 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fv 5454  df-dom 7103  df-siga 24483  df-sigagen 24514
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