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Theorem n0mmoeu 3285
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 1993 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 103   E.wex 1424    e. wcel 1436   E!weu 1945   E*wmo 1946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by: (None)
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