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Theorem elnev 27006
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 2767 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 df-v 2765 . . . . 5  |-  _V  =  { x  |  x  =  x }
32eqeq2i 2268 . . . 4  |-  ( { x  |  -.  x  =  A }  =  _V  <->  { x  |  -.  x  =  A }  =  {
x  |  x  =  x } )
4 equid 1818 . . . . . . 7  |-  x  =  x
54tbt 335 . . . . . 6  |-  ( -.  x  =  A  <->  ( -.  x  =  A  <->  x  =  x ) )
65albii 1554 . . . . 5  |-  ( A. x  -.  x  =  A  <->  A. x ( -.  x  =  A  <->  x  =  x
) )
7 alnex 1569 . . . . 5  |-  ( A. x  -.  x  =  A  <->  -.  E. x  x  =  A )
8 abbi 2368 . . . . 5  |-  ( A. x ( -.  x  =  A  <->  x  =  x
)  <->  { x  |  -.  x  =  A }  =  { x  |  x  =  x } )
96, 7, 83bitr3ri 269 . . . 4  |-  ( { x  |  -.  x  =  A }  =  {
x  |  x  =  x }  <->  -.  E. x  x  =  A )
103, 9bitri 242 . . 3  |-  ( { x  |  -.  x  =  A }  =  _V  <->  -. 
E. x  x  =  A )
1110necon2abii 2476 . 2  |-  ( E. x  x  =  A  <->  { x  |  -.  x  =  A }  =/=  _V )
121, 11bitri 242 1  |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2244    =/= wne 2421   _Vcvv 2763
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-ne 2423  df-v 2765
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