Theorem 0nep0 5301

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 5250	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4731	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2984	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2930  ∅c0 4283  {csn 4578

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 2706  ax-nul 5249

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 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-v 3440  df-dif 3902  df-nul 4284  df-sn 4579

This theorem is referenced by:  0inp0  5302  opthprc  5686  2dom  8965  pw2eng  9009  djuexb  9819  hashge3el3dif  14408  cat1  18019  isusp  24203  bj-1upln0  37153  clsk1indlem0  44224  mnuprdlem1  44455  mnuprdlem2  44456