Theorem 0nep0 5328

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 5277	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4752	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2986	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2932  ∅c0 4308  {csn 4601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-v 3461  df-dif 3929  df-nul 4309  df-sn 4602

This theorem is referenced by:  0inp0  5329  opthprc  5718  2dom  9044  pw2eng  9092  djuexb  9923  hashge3el3dif  14505  cat1  18110  isusp  24200  bj-1upln0  37027  clsk1indlem0  44065  mnuprdlem1  44296  mnuprdlem2  44297