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Theorem 0nep0 3945
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 3911 . . 3 ∅ ∈ V
21snnz 3514 . 2 {∅} ≠ ∅
32necomi 2305 1 ∅ ≠ {∅}
Colors of variables: wff set class
Syntax hints:  wne 2220  c0 3251  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2947  df-nul 3252  df-sn 3408
This theorem is referenced by:  0inp0  3946  opthprc  4418  2dom  6315
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