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Theorem elsigagen2 26759
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 988 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  A  e.  V )
21sgsiga 26753 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  (sigaGen `  A
)  e.  U. ran sigAlgebra )
3 sssigagen 26756 . . . 4  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
4 sspwb 4652 . . . . 5  |-  ( A 
C_  (sigaGen `  A )  <->  ~P A  C_  ~P (sigaGen `  A ) )
54biimpi 194 . . . 4  |-  ( A 
C_  (sigaGen `  A )  ->  ~P A  C_  ~P (sigaGen `  A ) )
61, 3, 53syl 20 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ~P A  C_ 
~P (sigaGen `  A )
)
7 simp2 989 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  C_  A )
8 simp3 990 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  ~<_  om )
9 ctex 26186 . . . . 5  |-  ( B  ~<_  om  ->  B  e.  _V )
10 elpwg 3979 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
118, 9, 103syl 20 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ( B  e.  ~P A  <->  B  C_  A
) )
127, 11mpbird 232 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P A )
136, 12sseldd 3468 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P (sigaGen `  A )
)
14 sigaclcu 26728 . 2  |-  ( ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  B  e. 
~P (sigaGen `  A )  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A ) )
152, 13, 8, 14syl3anc 1219 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    e. wcel 1758   _Vcvv 3078    C_ wss 3439   ~Pcpw 3971   U.cuni 4202   class class class wbr 4403   ran crn 4952   ` cfv 5529   omcom 6589    ~<_ cdom 7421  sigAlgebracsiga 26718  sigaGencsigagen 26749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fv 5537  df-dom 7425  df-siga 26719  df-sigagen 26750
This theorem is referenced by:  sxbrsigalem1  26867
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