Theorem 0nep0 5305

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 5254	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4735	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2987	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2933  ∅c0 4287  {csn 4582

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 2709  ax-nul 5253

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3444  df-dif 3906  df-nul 4288  df-sn 4583

This theorem is referenced by:  0inp0  5306  opthprc  5696  2dom  8979  pw2eng  9023  djuexb  9833  hashge3el3dif  14422  cat1  18033  isusp  24217  bj-1upln0  37257  clsk1indlem0  44397  mnuprdlem1  44628  mnuprdlem2  44629