MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0nep0 Structured version   Visualization version   GIF version

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
StepHypRef Expression
1 0ex 5269 . . 3 ∅ ∈ V
21snnz 4742 . 2 {∅} ≠ ∅
32necomi 2999 1 ∅ ≠ {∅}
Colors of variables: wff setvar class
Syntax hints:  wne 2944  c0 4287  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5268
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-v 3450  df-dif 3918  df-nul 4288  df-sn 4592
This theorem is referenced by:  0inp0  5319  opthprc  5701  2dom  8981  pw2eng  9029  djuexb  9852  hashge3el3dif  14392  cat1  17990  isusp  23629  bj-1upln0  35509  clsk1indlem0  42387  mnuprdlem1  42626  mnuprdlem2  42627
  Copyright terms: Public domain W3C validator