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Theorem 0nep0 4096
 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 4062 . . 3 ∅ ∈ V
21snnz 3649 . 2 {∅} ≠ ∅
32necomi 2394 1 ∅ ≠ {∅}
 Colors of variables: wff set class Syntax hints:   ≠ wne 2309  ∅c0 3367  {csn 3531 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4061 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3077  df-nul 3368  df-sn 3537 This theorem is referenced by:  0inp0  4097  opthprc  4597  2dom  6706  exmidpw  6809  exmidaclem  7080  pw1dom2  13359
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