Theorem 0nep0 5318

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 5260	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4733	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3009	1 ⊢ ∅ ≠ {∅}

This proof depends on syntax axioms:   ≠ wne 2955  ∅c0 4277  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4046  df-nul 4278  df-sn 4585

This theorem is used by:  0inp0  5319  opthprc  5714  2dom  9063  pw2eng  9107  djuexb  9995  hashge3el3dif  14637  cat1  18279  isusp  24587  kard0b  35885  bj-1upln0  38016  clsk1indlem0  45148  mnuprdlem1  45363  mnuprdlem2  45364