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Theorem n0moeu 4080
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 4074 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
21biimpi 206 . . 3 (𝐴 ≠ ∅ → ∃𝑥 𝑥𝐴)
32biantrurd 530 . 2 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴)))
4 eu5 2633 . 2 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∃*𝑥 𝑥𝐴))
53, 4syl6bbr 278 1 (𝐴 ≠ ∅ → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1853  wcel 2139  ∃!weu 2607  ∃*wmo 2608  wne 2932  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-nul 4059
This theorem is referenced by:  minveclem4a  23401
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