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Theorem abssexg 3962
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 3961 . 2  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 df-pw 3389 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
32eleq1i 2119 . . 3  |-  ( ~P A  e.  _V  <->  { x  |  x  C_  A }  e.  _V )
4 simpl 106 . . . . 5  |-  ( ( x  C_  A  /\  ph )  ->  x  C_  A
)
54ss2abi 3040 . . . 4  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  |  x  C_  A }
6 ssexg 3924 . . . 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 408 . . 3  |-  ( { x  |  x  C_  A }  e.  _V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
83, 7sylbi 118 . 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 101    e. wcel 1409   {cab 2042   _Vcvv 2574    C_ wss 2945   ~Pcpw 3387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389
This theorem is referenced by: (None)
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