Theorem 0nep0 5280

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 5231	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4712	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2998	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2943  ∅c0 4256  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-nul 4257  df-sn 4562

This theorem is referenced by:  0inp0  5281  opthprc  5651  2dom  8820  pw2eng  8865  djuexb  9667  hashge3el3dif  14200  cat1  17812  isusp  23413  bj-1upln0  35199  clsk1indlem0  41651  mnuprdlem1  41890  mnuprdlem2  41891