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Theorem 0inp0 4058
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 4057 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2296 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 167 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2304 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1314    =/= wne 2283   (/)c0 3331   {csn 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660  df-dif 3041  df-nul 3332  df-sn 3501
This theorem is referenced by: (None)
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