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Theorem 0inp0 4001
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 4000 . . 3 ∅ ≠ {∅}
2 neeq1 2268 . . 3 (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 166 . 2 (𝐴 = ∅ → 𝐴 ≠ {∅})
43neneqd 2276 1 (𝐴 = ∅ → ¬ 𝐴 = {∅})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wne 2255  c0 3286  {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-nul 3287  df-sn 3452
This theorem is referenced by: (None)
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