Theorem 0nep0 5358

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 5307	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4776	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2995	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2940  ∅c0 4333  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-nul 4334  df-sn 4627

This theorem is referenced by:  0inp0  5359  opthprc  5749  2dom  9070  pw2eng  9118  djuexb  9949  hashge3el3dif  14526  cat1  18142  isusp  24270  bj-1upln0  37010  clsk1indlem0  44054  mnuprdlem1  44291  mnuprdlem2  44292