Theorem n0mmoeu 3477

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 2102	1 |- ( E. x x e. A -> ( E* x x e. A <-> E! x x e. A ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1515   E!weu 2054   E*wmo 2055   e. wcel 2176

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058

This theorem is referenced by: (None)