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Theorem 0inp0 3999
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 3998 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2268 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 166 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2276 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    =/= wne 2255   (/)c0 3286   {csn 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3963
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-nul 3287  df-sn 3450
This theorem is referenced by: (None)
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