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Theorem compneOLD 27746
Description: Obsolete proof of compne 27745 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 1728 . . . . . 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 2283 . . . . . . . 8  |-  ( y  e.  { x  |  -.  x  e.  A } 
<->  [ y  /  x ]  -.  x  e.  A
)
4 df-clab 2283 . . . . . . . 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 2015 . . . . . . . 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 2290 . . . 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 27744 . . . . 5  |-  ( _V 
\  A )  =  { x  |  -.  x  e.  A }
1514eqcomi 2300 . . . 4  |-  { x  |  -.  x  e.  A }  =  ( _V  \  A )
16 abid2 2413 . . . 4  |-  { x  |  x  e.  A }  =  A
1715, 16eqeq12i 2309 . . 3  |-  ( { x  |  -.  x  e.  A }  =  {
x  |  x  e.  A }  <->  ( _V  \  A )  =  A )
1813, 17mtbi 289 . 2  |-  -.  ( _V  \  A )  =  A
19 df-ne 2461 . 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 1530    = wceq 1632   [wsb 1638    e. wcel 1696   {cab 2282    =/= wne 2459   _Vcvv 2801    \ cdif 3162
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-ne 2461  df-v 2803  df-dif 3168
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