Theorem 0nep0 5316

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 5265	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4743	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2980	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2926  ∅c0 4299  {csn 4592

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-nul 4300  df-sn 4593

This theorem is referenced by:  0inp0  5317  opthprc  5705  2dom  9004  pw2eng  9052  djuexb  9869  hashge3el3dif  14459  cat1  18066  isusp  24156  bj-1upln0  37004  clsk1indlem0  44037  mnuprdlem1  44268  mnuprdlem2  44269