Theorem 0nep0 5326

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 5267	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4739	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3013	1 ⊢ ∅ ≠ {∅}

This proof depends on syntax axioms:   ≠ wne 2959  ∅c0 4281  {csn 4588

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-nul 4282  df-sn 4589

This theorem is used by:  0inp0  5327  opthprc  5722  2dom  9096  pw2eng  9140  djuexb  10027  hashge3el3dif  14673  cat1  18322  isusp  24635  kard0b  35968  bj-1upln0  38137  clsk1indlem0  45302  mnuprdlem1  45517  mnuprdlem2  45518