Theorem 0nep0 5311

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 5254	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4732	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3010	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2956  ∅c0 4283  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5253

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3905  df-nul 4284  df-sn 4580

This theorem is referenced by:  0inp0  5312  opthprc  5707  2dom  9005  pw2eng  9049  djuexb  9861  hashge3el3dif  14494  cat1  18121  isusp  24309  bj-1upln0  37455  clsk1indlem0  44578  mnuprdlem1  44809  mnuprdlem2  44810