Theorem 0nep0 5275

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 5226	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4709	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2997	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2942  ∅c0 4253  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-nul 4254  df-sn 4559

This theorem is referenced by:  0inp0  5276  opthprc  5642  2dom  8774  pw2eng  8818  djuexb  9598  hashge3el3dif  14128  cat1  17728  isusp  23321  bj-1upln0  35126  clsk1indlem0  41540  mnuprdlem1  41779  mnuprdlem2  41780