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Theorem abssexg 4318
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 4317 . 2  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 df-pw 3737 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
32eleq1i 2443 . . 3  |-  ( ~P A  e.  _V  <->  { x  |  x  C_  A }  e.  _V )
4 simpl 444 . . . . 5  |-  ( ( x  C_  A  /\  ph )  ->  x  C_  A
)
54ss2abi 3351 . . . 4  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  |  x  C_  A }
6 ssexg 4283 . . . 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 652 . . 3  |-  ( { x  |  x  C_  A }  e.  _V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
83, 7sylbi 188 . 2  |-  ( ~P A  e.  _V  ->  { x  |  ( x 
C_  A  /\  ph ) }  e.  _V )
91, 8syl 16 1  |-  ( A  e.  V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   {cab 2366   _Vcvv 2892    C_ wss 3256   ~Pcpw 3735
This theorem is referenced by:  pmex  6952  tgval  16936
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-pow 4311
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-in 3263  df-ss 3270  df-pw 3737
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