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Theorem elsigagen2 23511
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  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A )
)

Proof of Theorem elsigagen2
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  A  e.  V )
21sgsiga 23505 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  (sigaGen `  A
)  e.  U. ran sigAlgebra )
3 sssigagen 23508 . . . 4  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
4 sspwb 4225 . . . . 5  |-  ( A 
C_  (sigaGen `  A )  <->  ~P A  C_  ~P (sigaGen `  A ) )
54biimpi 186 . . . 4  |-  ( A 
C_  (sigaGen `  A )  ->  ~P A  C_  ~P (sigaGen `  A ) )
61, 3, 53syl 18 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ~P A  C_ 
~P (sigaGen `  A )
)
7 simp2 956 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  C_  A )
8 simp3 957 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  ~<_  om )
9 ctex 23338 . . . . 5  |-  ( B  ~<_  om  ->  B  e.  _V )
10 elpwg 3634 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
118, 9, 103syl 18 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ( B  e.  ~P A  <->  B  C_  A
) )
127, 11mpbird 223 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P A )
136, 12sseldd 3183 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P (sigaGen `  A )
)
14 sigaclcu 23480 . 2  |-  ( ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  B  e. 
~P (sigaGen `  A )  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A ) )
152, 13, 8, 14syl3anc 1182 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1686   _Vcvv 2790    C_ wss 3154   ~Pcpw 3627   U.cuni 3829   class class class wbr 4025   omcom 4658   ran crn 4692   ` cfv 5257    ~<_ cdom 6863  sigAlgebracsiga 23470  sigaGencsigagen 23501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fv 5265  df-dom 6867  df-siga 23471  df-sigagen 23502
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