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Theorem 0inp0 3919
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 3918 . . 3 ∅ ≠ {∅}
2 neeq1 2218 . . 3 (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 157 . 2 (𝐴 = ∅ → 𝐴 ≠ {∅})
43neneqd 2226 1 (𝐴 = ∅ → ¬ 𝐴 = {∅})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1243  wne 2204  c0 3224  {csn 3375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-nul 3225  df-sn 3381
This theorem is referenced by: (None)
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