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Theorem bnj521 29081
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj521  |-  ( A  i^i  { A }
)  =  (/)

Proof of Theorem bnj521
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elirr 7328 . . . 4  |-  -.  A  e.  A
2 elin 3371 . . . . . 6  |-  ( x  e.  ( A  i^i  { A } )  <->  ( x  e.  A  /\  x  e.  { A } ) )
3 elsn 3668 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
4 eleq1 2356 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
54biimpac 472 . . . . . . 7  |-  ( ( x  e.  A  /\  x  =  A )  ->  A  e.  A )
63, 5sylan2b 461 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  { A } )  ->  A  e.  A )
72, 6sylbi 187 . . . . 5  |-  ( x  e.  ( A  i^i  { A } )  ->  A  e.  A )
87exlimiv 1624 . . . 4  |-  ( E. x  x  e.  ( A  i^i  { A } )  ->  A  e.  A )
91, 8mto 167 . . 3  |-  -.  E. x  x  e.  ( A  i^i  { A }
)
10 n0 3477 . . 3  |-  ( ( A  i^i  { A } )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  { A } ) )
119, 10mtbir 290 . 2  |-  -.  ( A  i^i  { A }
)  =/=  (/)
12 nne 2463 . 2  |-  ( -.  ( A  i^i  { A } )  =/=  (/)  <->  ( A  i^i  { A } )  =  (/) )
1311, 12mpbi 199 1  |-  ( A  i^i  { A }
)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164   (/)c0 3468   {csn 3653
This theorem is referenced by:  bnj927  29116  bnj535  29238
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
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-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-nul 3469  df-sn 3659  df-pr 3660
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