Theorem 0nep0 5364

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 5313	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4781	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2993	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2938  ∅c0 4339  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-nul 4340  df-sn 4632

This theorem is referenced by:  0inp0  5365  opthprc  5753  2dom  9069  pw2eng  9117  djuexb  9947  hashge3el3dif  14523  cat1  18151  isusp  24286  bj-1upln0  36992  clsk1indlem0  44031  mnuprdlem1  44268  mnuprdlem2  44269