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Theorem 0inp0 5312
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 5311 . . 3 ∅ ≠ {∅}
2 neeq1 3004 . . 3 (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 257 . 2 (𝐴 = ∅ → 𝐴 ≠ {∅})
43neneqd 2946 1 (𝐴 = ∅ → ¬ 𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1541  wne 2941  c0 4280  {csn 4584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-v 3445  df-dif 3911  df-nul 4281  df-sn 4585
This theorem is referenced by:  eqsnuniex  5314  dtruALT  5341  zfpair  5374
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