ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  abssexg Unicode version

Theorem abssexg 4074
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 4072 . 2  |-  ( A  e.  V  ->  ~P A  e.  _V )
2 df-pw 3480 . . . 4  |-  ~P A  =  { x  |  x 
C_  A }
32eleq1i 2181 . . 3  |-  ( ~P A  e.  _V  <->  { x  |  x  C_  A }  e.  _V )
4 simpl 108 . . . . 5  |-  ( ( x  C_  A  /\  ph )  ->  x  C_  A
)
54ss2abi 3137 . . . 4  |-  { x  |  ( x  C_  A  /\  ph ) } 
C_  { x  |  x  C_  A }
6 ssexg 4035 . . . 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 418 . . 3  |-  ( { x  |  x  C_  A }  e.  _V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
83, 7sylbi 120 . 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 103    e. wcel 1463   {cab 2101   _Vcvv 2658    C_ wss 3039   ~Pcpw 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-pw 3480
This theorem is referenced by:  pmex  6513  tgval  12113
  Copyright terms: Public domain W3C validator