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Theorem bnj521 29041
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 7558 . . . 4  |-  -.  A  e.  A
2 elin 3522 . . . . . 6  |-  ( x  e.  ( A  i^i  { A } )  <->  ( x  e.  A  /\  x  e.  { A } ) )
3 elsn 3821 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
4 eleq1 2495 . . . . . . . 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 1644 . . . 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 3629 . . 3  |-  ( ( A  i^i  { A } )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  { A } ) )
119, 10mtbir 291 . 2  |-  -.  ( A  i^i  { A }
)  =/=  (/)
12 nne 2602 . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311   (/)c0 3620   {csn 3806
This theorem is referenced by:  bnj927  29076  bnj535  29198
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-reg 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-nul 3621  df-sn 3812  df-pr 3813
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