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Theorem 0inp0 4942
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 4941 . . 3 ∅ ≠ {∅}
2 neeq1 2958 . . 3 (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 248 . 2 (𝐴 = ∅ → 𝐴 ≠ {∅})
43neneqd 2901 1 (𝐴 = ∅ → ¬ 𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1596  wne 2896  c0 4023  {csn 4285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-nul 4897
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-v 3306  df-dif 3683  df-nul 4024  df-sn 4286
This theorem is referenced by:  dtruALT  5004  zfpair  5009  dtruALT2  5016
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