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Theorem abssexg 4622
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 4621 . 2  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 df-pw 4001 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
32eleq1i 2531 . . 3  |-  ( ~P A  e.  _V  <->  { x  |  x  C_  A }  e.  _V )
4 simpl 455 . . . . 5  |-  ( ( x  C_  A  /\  ph )  ->  x  C_  A
)
54ss2abi 3558 . . . 4  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  |  x  C_  A }
6 ssexg 4583 . . . 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 668 . . 3  |-  ( { x  |  x  C_  A }  e.  _V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
83, 7sylbi 195 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   {cab 2439   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-pow 4615
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001
This theorem is referenced by:  pmex  7417  tgval  19623
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