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Theorem compneOLD 27643
Description: Obsolete proof of compne 27642 as of 28-Jun-2015. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
compneOLD  |-  ( _V 
\  A )  =/= 
A

Proof of Theorem compneOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm5.19 349 . . . . 5  |-  -.  ( [ y  /  x ] x  e.  A  <->  -. 
[ y  /  x ] x  e.  A
)
2 sp 1716 . . . . . 6  |-  ( A. y ( y  e. 
{ x  |  -.  x  e.  A }  <->  y  e.  { x  |  x  e.  A }
)  ->  ( y  e.  { x  |  -.  x  e.  A }  <->  y  e.  { x  |  x  e.  A }
) )
3 df-clab 2270 . . . . . . . 8  |-  ( y  e.  { x  |  -.  x  e.  A } 
<->  [ y  /  x ]  -.  x  e.  A
)
4 df-clab 2270 . . . . . . . 8  |-  ( y  e.  { x  |  x  e.  A }  <->  [ y  /  x ]
x  e.  A )
53, 4bibi12i 306 . . . . . . 7  |-  ( ( y  e.  { x  |  -.  x  e.  A } 
<->  y  e.  { x  |  x  e.  A } )  <->  ( [
y  /  x ]  -.  x  e.  A  <->  [ y  /  x ]
x  e.  A ) )
6 sbn 2002 . . . . . . . 8  |-  ( [ y  /  x ]  -.  x  e.  A  <->  -. 
[ y  /  x ] x  e.  A
)
76bibi1i 305 . . . . . . 7  |-  ( ( [ y  /  x ]  -.  x  e.  A  <->  [ y  /  x ]
x  e.  A )  <-> 
( -.  [ y  /  x ] x  e.  A  <->  [ y  /  x ] x  e.  A
) )
8 bicom 191 . . . . . . 7  |-  ( ( -.  [ y  /  x ] x  e.  A  <->  [ y  /  x ]
x  e.  A )  <-> 
( [ y  /  x ] x  e.  A  <->  -. 
[ y  /  x ] x  e.  A
) )
95, 7, 83bitri 262 . . . . . 6  |-  ( ( y  e.  { x  |  -.  x  e.  A } 
<->  y  e.  { x  |  x  e.  A } )  <->  ( [
y  /  x ]
x  e.  A  <->  -.  [ y  /  x ] x  e.  A ) )
102, 9sylib 188 . . . . 5  |-  ( A. y ( y  e. 
{ x  |  -.  x  e.  A }  <->  y  e.  { x  |  x  e.  A }
)  ->  ( [
y  /  x ]
x  e.  A  <->  -.  [ y  /  x ] x  e.  A ) )
111, 10mto 167 . . . 4  |-  -.  A. y ( y  e. 
{ x  |  -.  x  e.  A }  <->  y  e.  { x  |  x  e.  A }
)
12 dfcleq 2277 . . . 4  |-  ( { x  |  -.  x  e.  A }  =  {
x  |  x  e.  A }  <->  A. y
( y  e.  {
x  |  -.  x  e.  A }  <->  y  e.  { x  |  x  e.  A } ) )
1311, 12mtbir 290 . . 3  |-  -.  {
x  |  -.  x  e.  A }  =  {
x  |  x  e.  A }
14 compeq 27641 . . . . 5  |-  ( _V 
\  A )  =  { x  |  -.  x  e.  A }
1514eqcomi 2287 . . . 4  |-  { x  |  -.  x  e.  A }  =  ( _V  \  A )
16 abid2 2400 . . . 4  |-  { x  |  x  e.  A }  =  A
1715, 16eqeq12i 2296 . . 3  |-  ( { x  |  -.  x  e.  A }  =  {
x  |  x  e.  A }  <->  ( _V  \  A )  =  A )
1813, 17mtbi 289 . 2  |-  -.  ( _V  \  A )  =  A
19 df-ne 2448 . 2  |-  ( ( _V  \  A )  =/=  A  <->  -.  ( _V  \  A )  =  A )
2018, 19mpbir 200 1  |-  ( _V 
\  A )  =/= 
A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1527    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269    =/= wne 2446   _Vcvv 2788    \ cdif 3149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155
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