Theorem 0nep0 5299

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

Step	Hyp	Ref	Expression
1		0ex 5242	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4720	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2986	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2932  ∅c0 4273  {csn 4567

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3431  df-dif 3892  df-nul 4274  df-sn 4568

This theorem is referenced by:  0inp0  5300  opthprc  5695  2dom  8977  pw2eng  9021  djuexb  9833  hashge3el3dif  14449  cat1  18064  isusp  24226  bj-1upln0  37316  clsk1indlem0  44468  mnuprdlem1  44699  mnuprdlem2  44700