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 5270	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4742	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3012	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2958  ∅c0 4286  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-nul 4287  df-sn 4590

This theorem is referenced by:  0inp0  5329  opthprc  5725  2dom  9023  pw2eng  9067  djuexb  9891  hashge3el3dif  14520  cat1  18149  isusp  24418  kard0b  35572  bj-1upln0  37645  clsk1indlem0  44767  mnuprdlem1  44982  mnuprdlem2  44983