Theorem n0mmoeu 3464

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 2090	1 |- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1503   E!weu 2042   E*wmo 2043   e. wcel 2164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046

This theorem is referenced by: (None)