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Theorem abssexg 4008
Description: Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
abssexg  |-  ( A  e.  V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem abssexg
StepHypRef Expression
1 pwexg 4007 . 2  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 df-pw 3427 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
32eleq1i 2153 . . 3  |-  ( ~P A  e.  _V  <->  { x  |  x  C_  A }  e.  _V )
4 simpl 107 . . . . 5  |-  ( ( x  C_  A  /\  ph )  ->  x  C_  A
)
54ss2abi 3091 . . . 4  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  |  x  C_  A }
6 ssexg 3970 . . . 4  |-  ( ( { x  |  ( x  C_  A  /\  ph ) }  C_  { x  |  x  C_  A }  /\  { x  |  x 
C_  A }  e.  _V )  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
75, 6mpan 415 . . 3  |-  ( { x  |  x  C_  A }  e.  _V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
83, 7sylbi 119 . 2  |-  ( ~P A  e.  _V  ->  { x  |  ( x 
C_  A  /\  ph ) }  e.  _V )
91, 8syl 14 1  |-  ( A  e.  V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   {cab 2074   _Vcvv 2619    C_ wss 2997   ~Pcpw 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  pmex  6390
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