Theorem 0nep0 5297

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 5246	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4728	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2979	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2925  ∅c0 4284  {csn 4577

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5245

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3438  df-dif 3906  df-nul 4285  df-sn 4578

This theorem is referenced by:  0inp0  5298  opthprc  5683  2dom  8955  pw2eng  9000  djuexb  9805  hashge3el3dif  14394  cat1  18004  isusp  24147  bj-1upln0  36983  clsk1indlem0  44014  mnuprdlem1  44245  mnuprdlem2  44246