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Theorem elnev 31247
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 3122 . 2  |-  ( A  e.  _V  <->  E. x  x  =  A )
2 df-v 3120 . . . . 5  |-  _V  =  { x  |  x  =  x }
32eqeq2i 2485 . . . 4  |-  ( { x  |  -.  x  =  A }  =  _V  <->  { x  |  -.  x  =  A }  =  {
x  |  x  =  x } )
4 equid 1740 . . . . . . 7  |-  x  =  x
54tbt 344 . . . . . 6  |-  ( -.  x  =  A  <->  ( -.  x  =  A  <->  x  =  x ) )
65albii 1620 . . . . 5  |-  ( A. x  -.  x  =  A  <->  A. x ( -.  x  =  A  <->  x  =  x
) )
7 alnex 1598 . . . . 5  |-  ( A. x  -.  x  =  A  <->  -.  E. x  x  =  A )
8 abbi 2598 . . . . 5  |-  ( A. x ( -.  x  =  A  <->  x  =  x
)  <->  { x  |  -.  x  =  A }  =  { x  |  x  =  x } )
96, 7, 83bitr3ri 276 . . . 4  |-  ( { x  |  -.  x  =  A }  =  {
x  |  x  =  x }  <->  -.  E. x  x  =  A )
103, 9bitri 249 . . 3  |-  ( { x  |  -.  x  =  A }  =  _V  <->  -. 
E. x  x  =  A )
1110necon2abii 2733 . 2  |-  ( E. x  x  =  A  <->  { x  |  -.  x  =  A }  =/=  _V )
121, 11bitri 249 1  |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   _Vcvv 3118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ne 2664  df-v 3120
This theorem is referenced by: (None)
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