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Theorem bnj521 34212
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 8016 . . . 4  |-  -.  A  e.  A
2 elin 3673 . . . . . 6  |-  ( x  e.  ( A  i^i  { A } )  <->  ( x  e.  A  /\  x  e.  { A } ) )
3 elsn 4030 . . . . . . 7  |-  ( x  e.  { A }  <->  x  =  A )
4 eleq1 2526 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
54biimpac 484 . . . . . . 7  |-  ( ( x  e.  A  /\  x  =  A )  ->  A  e.  A )
63, 5sylan2b 473 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  { A } )  ->  A  e.  A )
72, 6sylbi 195 . . . . 5  |-  ( x  e.  ( A  i^i  { A } )  ->  A  e.  A )
87exlimiv 1727 . . . 4  |-  ( E. x  x  e.  ( A  i^i  { A } )  ->  A  e.  A )
91, 8mto 176 . . 3  |-  -.  E. x  x  e.  ( A  i^i  { A }
)
10 n0 3793 . . 3  |-  ( ( A  i^i  { A } )  =/=  (/)  <->  E. x  x  e.  ( A  i^i  { A } ) )
119, 10mtbir 297 . 2  |-  -.  ( A  i^i  { A }
)  =/=  (/)
12 nne 2655 . 2  |-  ( -.  ( A  i^i  { A } )  =/=  (/)  <->  ( A  i^i  { A } )  =  (/) )
1311, 12mpbi 208 1  |-  ( A  i^i  { A }
)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649    i^i cin 3460   (/)c0 3783   {csn 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-reg 8010
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-nul 3784  df-sn 4017  df-pr 4019
This theorem is referenced by:  bnj927  34247  bnj535  34368
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