Theorem n0mmoeu 3345

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 2023	1 |- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1451   e. wcel 1463   E!weu 1975   E*wmo 1976

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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979

This theorem is referenced by: (None)