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Theorem 0nep0 5142
 Description: The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
0nep0 ∅ ≠ {∅}

Proof of Theorem 0nep0
StepHypRef Expression
1 0ex 5096 . . 3 ∅ ∈ V
21snnz 4612 . 2 {∅} ≠ ∅
32necomi 3036 1 ∅ ≠ {∅}
 Colors of variables: wff setvar class Syntax hints:   ≠ wne 2982  ∅c0 4206  {csn 4466 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767  ax-nul 5095 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-v 3434  df-dif 3857  df-nul 4207  df-sn 4467 This theorem is referenced by:  0inp0  5143  opthprc  5494  2dom  8420  pw2eng  8460  djuexb  9173  hashge3el3dif  13678  isusp  22541  bj-1upln0  33872  clsk1indlem0  39827  mnuprdlem1  40057  mnuprdlem2  40058
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