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Theorem 0nep0 5355
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 5306 . . 3 ∅ ∈ V
21snnz 4779 . 2 {∅} ≠ ∅
32necomi 2995 1 ∅ ≠ {∅}
Colors of variables: wff setvar class
Syntax hints:  wne 2940  c0 4321  {csn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3950  df-nul 4322  df-sn 4628
This theorem is referenced by:  0inp0  5356  opthprc  5738  2dom  9026  pw2eng  9074  djuexb  9900  hashge3el3dif  14443  cat1  18043  isusp  23757  bj-1upln0  35878  clsk1indlem0  42777  mnuprdlem1  43016  mnuprdlem2  43017
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