Theorem 0nep0 5286

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 5229	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4708	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2988	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2934  ∅c0 4261  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-v 3433  df-dif 3886  df-nul 4262  df-sn 4556

This theorem is referenced by:  0inp0  5287  opthprc  5682  2dom  8967  pw2eng  9011  djuexb  9824  hashge3el3dif  14440  cat1  18055  isusp  24244  bj-1upln0  37362  clsk1indlem0  44485  mnuprdlem1  44716  mnuprdlem2  44717