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Theorem notab 3451
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 2565 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
2 rabab 2818 . . 3  |-  { x  e.  _V  |  -.  ph }  =  { x  |  -.  ph }
31, 2eqtr3i 2318 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  { x  |  -.  ph }
4 difab 3450 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  { x  |  ( x  e. 
_V  /\  -.  ph ) }
5 abid2 2413 . . . 4  |-  { x  |  x  e.  _V }  =  _V
65difeq1i 3303 . . 3  |-  ( { x  |  x  e. 
_V }  \  {
x  |  ph }
)  =  ( _V 
\  { x  | 
ph } )
74, 6eqtr3i 2318 . 2  |-  { x  |  ( x  e. 
_V  /\  -.  ph ) }  =  ( _V  \  { x  |  ph } )
83, 7eqtr3i 2318 1  |-  { x  |  -.  ph }  =  ( _V  \  { x  |  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   _Vcvv 2801    \ cdif 3162
This theorem is referenced by:  dfif3  3588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168
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