Theorem 0nep0 5326

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 5269	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4744	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3018	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2964  ∅c0 4294  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-nul 4295  df-sn 4592

This theorem is referenced by:  0inp0  5327  opthprc  5723  2dom  9023  pw2eng  9067  djuexb  9891  hashge3el3dif  14520  cat1  18150  isusp  24383  bj-1upln0  37529  clsk1indlem0  44654  mnuprdlem1  44869  mnuprdlem2  44870