Theorem n0mmoeu 3425

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

Step	Hyp	Ref	Expression
1		exmoeu2 2062	1 |- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1480   E!weu 2014   E*wmo 2015   e. wcel 2136

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by: (None)