Theorem n0mmoeu 3437

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 2072	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1490  ∃!weu 2024  ∃*wmo 2025   ∈ wcel 2146

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028

This theorem is referenced by: (None)