Theorem 0nep0 4762

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 4718	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4252	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2836	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2780  ∅c0 3874  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-nul 3875  df-sn 4126

This theorem is referenced by:  0inp0  4763  opthprc  5089  2dom  7915  pw2eng  7951  hashge3el3dif  13122  isusp  21875  bj-1upln0  32190  clsk1indlem0  37359