Theorem 0nep0 5291

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 5240	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4724	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2982	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2928  ∅c0 4278  {csn 4571

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5239

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3900  df-nul 4279  df-sn 4572

This theorem is referenced by:  0inp0  5292  opthprc  5675  2dom  8947  pw2eng  8991  djuexb  9797  hashge3el3dif  14389  cat1  17999  isusp  24171  bj-1upln0  37043  clsk1indlem0  44074  mnuprdlem1  44305  mnuprdlem2  44306