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Theorem 0inp0 4573
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)
Assertion
Ref Expression
0inp0  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )

Proof of Theorem 0inp0
StepHypRef Expression
1 0nep0 4572 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2705 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 241 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2648 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1452    =/= wne 2641   (/)c0 3722   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-nul 3723  df-sn 3960
This theorem is referenced by:  dtruALT  4632  zfpair  4637  dtruALT2  4644
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