MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0moeu Structured version   Visualization version   GIF version

Theorem n0moeu 4357
Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)
Assertion
Ref Expression
n0moeu (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem n0moeu
StepHypRef Expression
1 n0 4347 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 215 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32biantrurd 532 . 2 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴)))
4 df-eu 2559 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴))
53, 4bitr4di 289 1 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1774  wcel 2099  ∃*wmo 2528  ∃!weu 2558  wne 2937  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-eu 2559  df-clab 2706  df-cleq 2720  df-ne 2938  df-dif 3950  df-nul 4324
This theorem is referenced by:  minveclem4a  25371
  Copyright terms: Public domain W3C validator