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Theorem 0inp0 4127
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 4126 . . 3 ∅ ≠ {∅}
2 neeq1 2340 . . 3 (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 167 . 2 (𝐴 = ∅ → 𝐴 ≠ {∅})
43neneqd 2348 1 (𝐴 = ∅ → ¬ 𝐴 = {∅})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1335  wne 2327  c0 3394  {csn 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-nul 3395  df-sn 3566
This theorem is referenced by: (None)
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