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Theorem bnj521 28442
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 7499 . . . 4  |-  -.  A  e.  A
2 elin 3473 . . . . . 6  |-  ( x  e.  ( A  i^i  { A } )  <->  ( x  e.  A  /\  x  e.  { A } ) )
3 elsn 3772 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
4 eleq1 2447 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
54biimpac 473 . . . . . . 7  |-  ( ( x  e.  A  /\  x  =  A )  ->  A  e.  A )
63, 5sylan2b 462 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  { A } )  ->  A  e.  A )
72, 6sylbi 188 . . . . 5  |-  ( x  e.  ( A  i^i  { A } )  ->  A  e.  A )
87exlimiv 1641 . . . 4  |-  ( E. x  x  e.  ( A  i^i  { A } )  ->  A  e.  A )
91, 8mto 169 . . 3  |-  -.  E. x  x  e.  ( A  i^i  { A }
)
10 n0 3580 . . 3  |-  ( ( A  i^i  { A } )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  { A } ) )
119, 10mtbir 291 . 2  |-  -.  ( A  i^i  { A }
)  =/=  (/)
12 nne 2554 . 2  |-  ( -.  ( A  i^i  { A } )  =/=  (/)  <->  ( A  i^i  { A } )  =  (/) )
1311, 12mpbi 200 1  |-  ( A  i^i  { A }
)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550    i^i cin 3262   (/)c0 3571   {csn 3757
This theorem is referenced by:  bnj927  28477  bnj535  28599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-reg 7493
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-nul 3572  df-sn 3763  df-pr 3764
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