Theorem 0nep0 5319

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 5261	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4737	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3010	1 ⊢ ∅ ≠ {∅}

This proof depends on syntax axioms:   ≠ wne 2956  ∅c0 4279  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3453  df-dif 3902  df-nul 4280  df-sn 4585

This theorem is used by:  0inp0  5320  opthprc  5715  2dom  9051  pw2eng  9095  djuexb  9983  hashge3el3dif  14625  cat1  18265  isusp  24573  kard0b  35810  bj-1upln0  37902  clsk1indlem0  45026  mnuprdlem1  45241  mnuprdlem2  45242