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Theorem bnj521 28435
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 7492 . . . 4  |-  -.  A  e.  A
2 elin 3466 . . . . . 6  |-  ( x  e.  ( A  i^i  { A } )  <->  ( x  e.  A  /\  x  e.  { A } ) )
3 elsn 3765 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
4 eleq1 2440 . . . . . . . 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 3573 . . 3  |-  ( ( A  i^i  { A } )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  { A } ) )
119, 10mtbir 291 . 2  |-  -.  ( A  i^i  { A }
)  =/=  (/)
12 nne 2547 . 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 2543    i^i cin 3255   (/)c0 3564   {csn 3750
This theorem is referenced by:  bnj927  28470  bnj535  28592
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-reg 7486
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 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-nul 3565  df-sn 3756  df-pr 3757
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