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Theorem notab 3267
Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
Assertion
Ref Expression
notab  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )

Proof of Theorem notab
StepHypRef Expression
1 df-rab 2368 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
2 rabab 2640 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  -.  ph }
31, 2eqtr3i 2110 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  { x  |  -.  ph }
4 difab 3266 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
5 abid2 2208 . . . 4  |-  { x  |  x  e.  _V }  =  _V
65difeq1i 3112 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  ( _V 
\  { x  | 
ph } )
74, 6eqtr3i 2110 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  ( _V  \  { x  |  ph } )
83, 7eqtr3i 2110 1  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   {crab 2363   _Vcvv 2619    \ cdif 2994
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-in1 579  ax-in2 580  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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 2999
This theorem is referenced by:  dfif3  3402
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