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Theorem n0mmoeu 3326
 Description: A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)
Assertion
Ref Expression
n0mmoeu (∃𝑥 𝑥𝐴 → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem n0mmoeu
StepHypRef Expression
1 exmoeu2 2008 1 (∃𝑥 𝑥𝐴 → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∃wex 1436   ∈ wcel 1448  ∃!weu 1960  ∃*wmo 1961 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483 This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964 This theorem is referenced by: (None)
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