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Theorem n0mmoeu 3431
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  |-  ( E. x  x  e.  A  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A )
)
Distinct variable group:    x, A

Proof of Theorem n0mmoeu
StepHypRef Expression
1 exmoeu2 2067 1  |-  ( E. x  x  e.  A  ->  ( E* x  x  e.  A  <->  E! x  x  e.  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   E.wex 1485   E!weu 2019   E*wmo 2020    e. wcel 2141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023
This theorem is referenced by: (None)
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