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Theorem n0moeu 4317
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 4307 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 215 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32biantrurd 534 . 2 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴)))
4 df-eu 2564 . 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 397  wex 1782  wcel 2107  ∃*wmo 2533  ∃!weu 2563  wne 2940  c0 4283
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-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-eu 2564  df-clab 2711  df-cleq 2725  df-ne 2941  df-dif 3914  df-nul 4284
This theorem is referenced by:  minveclem4a  24810
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