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Theorem elnev 36441
Description: Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
Assertion
Ref Expression
elnev  |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
Distinct variable group:    x, A

Proof of Theorem elnev
StepHypRef Expression
1 isset 3091 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 df-v 3089 . . . . 5  |-  _V  =  { x  |  x  =  x }
32eqeq2i 2447 . . . 4  |-  ( { x  |  -.  x  =  A }  =  _V  <->  { x  |  -.  x  =  A }  =  {
x  |  x  =  x } )
4 equid 1842 . . . . . . 7  |-  x  =  x
54tbt 345 . . . . . 6  |-  ( -.  x  =  A  <->  ( -.  x  =  A  <->  x  =  x ) )
65albii 1687 . . . . 5  |-  ( A. x  -.  x  =  A  <->  A. x ( -.  x  =  A  <->  x  =  x
) )
7 alnex 1661 . . . . 5  |-  ( A. x  -.  x  =  A  <->  -.  E. x  x  =  A )
8 abbi 2560 . . . . 5  |-  ( A. x ( -.  x  =  A  <->  x  =  x
)  <->  { x  |  -.  x  =  A }  =  { x  |  x  =  x } )
96, 7, 83bitr3ri 279 . . . 4  |-  ( { x  |  -.  x  =  A }  =  {
x  |  x  =  x }  <->  -.  E. x  x  =  A )
103, 9bitri 252 . . 3  |-  ( { x  |  -.  x  =  A }  =  _V  <->  -. 
E. x  x  =  A )
1110necon2abii 2697 . 2  |-  ( E. x  x  =  A  <->  { x  |  -.  x  =  A }  =/=  _V )
121, 11bitri 252 1  |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414    =/= wne 2625   _Vcvv 3087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-ne 2627  df-v 3089
This theorem is referenced by: (None)
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