Theorem 0nep0 5376

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 5325	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4801	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3001	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2946  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-nul 4353  df-sn 4649

This theorem is referenced by:  0inp0  5377  opthprc  5764  2dom  9095  pw2eng  9144  djuexb  9978  hashge3el3dif  14536  cat1  18164  isusp  24291  bj-1upln0  36975  clsk1indlem0  44003  mnuprdlem1  44241  mnuprdlem2  44242