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Theorem 0inp0 4198
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 4197 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2467 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 224 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2475 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    =/= wne 2459   (/)c0 3468   {csn 3653
This theorem is referenced by:  dtruALT  4223  zfpair  4228  dtruALT2  4235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-nul 3469  df-sn 3659
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