Theorem 0nep0 4866

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 4823	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4340	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2877	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2823  ∅c0 3948  {csn 4210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-nul 3949  df-sn 4211

This theorem is referenced by:  0inp0  4867  opthprc  5201  2dom  8070  pw2eng  8107  hashge3el3dif  13306  isusp  22112  bj-1upln0  33122  clsk1indlem0  38656