Theorem 0nep0 5356

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 5307	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4780	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2995	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2940  ∅c0 4322  {csn 4628

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 5306

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 3951  df-nul 4323  df-sn 4629

This theorem is referenced by:  0inp0  5357  opthprc  5740  2dom  9032  pw2eng  9080  djuexb  9906  hashge3el3dif  14451  cat1  18051  isusp  23986  bj-1upln0  36193  clsk1indlem0  43094  mnuprdlem1  43333  mnuprdlem2  43334