Theorem n0mmoeu 3476

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	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem n0mmoeu

Step	Hyp	Ref	Expression
1		exmoeu2 2101	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1514  ∃!weu 2053  ∃*wmo 2054   ∈ wcel 2175

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057

This theorem is referenced by: (None)