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Theorem 0inp0 4219
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 4218 . . 3  |-  (/)  =/=  { (/)
}
2 neeq1 2487 . . 3  |-  ( A  =  (/)  ->  ( A  =/=  { (/) }  <->  (/)  =/=  { (/)
} ) )
31, 2mpbiri 224 . 2  |-  ( A  =  (/)  ->  A  =/= 
{ (/) } )
43neneqd 2495 1  |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1633    =/= wne 2479   (/)c0 3489   {csn 3674
This theorem is referenced by:  dtruALT  4244  zfpair  4249  dtruALT2  4256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-v 2824  df-dif 3189  df-nul 3490  df-sn 3680
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