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Theorem n0mmoeu 3410
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 2054 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 1472   E!weu 2006   E*wmo 2007    e. wcel 2128
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010
This theorem is referenced by: (None)
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