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Theorem 0inp0 4374
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
Assertion
Ref Expression
0inp0  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )

Proof of Theorem 0inp0
StepHypRef Expression
1 0nep0 4373 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2611 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 226 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2619 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    =/= wne 2601   (/)c0 3630   {csn 3816
This theorem is referenced by:  dtruALT  4399  zfpair  4404  dtruALT2  4411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-nul 3631  df-sn 3822
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