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Theorem elnev 27615
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 2960 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 df-v 2958 . . . . 5  |-  _V  =  { x  |  x  =  x }
32eqeq2i 2446 . . . 4  |-  ( { x  |  -.  x  =  A }  =  _V  <->  { x  |  -.  x  =  A }  =  {
x  |  x  =  x } )
4 equid 1688 . . . . . . 7  |-  x  =  x
54tbt 334 . . . . . 6  |-  ( -.  x  =  A  <->  ( -.  x  =  A  <->  x  =  x ) )
65albii 1575 . . . . 5  |-  ( A. x  -.  x  =  A  <->  A. x ( -.  x  =  A  <->  x  =  x
) )
7 alnex 1552 . . . . 5  |-  ( A. x  -.  x  =  A  <->  -.  E. x  x  =  A )
8 abbi 2546 . . . . 5  |-  ( A. x ( -.  x  =  A  <->  x  =  x
)  <->  { x  |  -.  x  =  A }  =  { x  |  x  =  x } )
96, 7, 83bitr3ri 268 . . . 4  |-  ( { x  |  -.  x  =  A }  =  {
x  |  x  =  x }  <->  -.  E. x  x  =  A )
103, 9bitri 241 . . 3  |-  ( { x  |  -.  x  =  A }  =  _V  <->  -. 
E. x  x  =  A )
1110necon2abii 2659 . 2  |-  ( E. x  x  =  A  <->  { x  |  -.  x  =  A }  =/=  _V )
121, 11bitri 241 1  |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   _Vcvv 2956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-ne 2601  df-v 2958
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