MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0inp0 Structured version   Visualization version   GIF version

Theorem 0inp0 5359
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)
Assertion
Ref Expression
0inp0 (𝐴 = ∅ → ¬ 𝐴 = {∅})

Proof of Theorem 0inp0
StepHypRef Expression
1 0nep0 5358 . . 3 ∅ ≠ {∅}
2 neeq1 3003 . . 3 (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 258 . 2 (𝐴 = ∅ → 𝐴 ≠ {∅})
43neneqd 2945 1 (𝐴 = ∅ → ¬ 𝐴 = {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wne 2940  c0 4333  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-nul 4334  df-sn 4627
This theorem is referenced by:  eqsnuniex  5361  dtruALT  5388  zfpair  5421
  Copyright terms: Public domain W3C validator