Theorem 0nep0 5303

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 5252	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4733	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2986	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2932  ∅c0 4285  {csn 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3442  df-dif 3904  df-nul 4286  df-sn 4581

This theorem is referenced by:  0inp0  5304  opthprc  5688  2dom  8967  pw2eng  9011  djuexb  9821  hashge3el3dif  14410  cat1  18021  isusp  24205  bj-1upln0  37210  clsk1indlem0  44282  mnuprdlem1  44513  mnuprdlem2  44514