Theorem 0nep0 5353

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 5302	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4777	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2991	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2936  ∅c0 4319  {csn 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5301

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-nul 4320  df-sn 4626

This theorem is referenced by:  0inp0  5354  opthprc  5737  2dom  9049  pw2eng  9097  djuexb  9927  hashge3el3dif  14474  cat1  18080  isusp  24160  bj-1upln0  36483  clsk1indlem0  43462  mnuprdlem1  43700  mnuprdlem2  43701