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Theorem n0moeu 4245
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 4235 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 219 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32biantrurd 536 . 2 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴)))
4 df-eu 2570 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴))
53, 4bitr4di 292 1 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1786  wcel 2114  ∃*wmo 2538  ∃!weu 2569  wne 2934  c0 4211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-eu 2570  df-clab 2717  df-cleq 2730  df-ne 2935  df-dif 3846  df-nul 4212
This theorem is referenced by:  minveclem4a  24182
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