Metamath Proof Explorer


Theorem relsn2

Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022)

Ref Expression
Assertion relsn2 ( 𝐴𝑉 → ( Rel { 𝐴 } ↔ dom { 𝐴 } ≠ ∅ ) )

Proof

Step Hyp Ref Expression
1 relsng ( 𝐴𝑉 → ( Rel { 𝐴 } ↔ 𝐴 ∈ ( V × V ) ) )
2 dmsnn0 ( 𝐴 ∈ ( V × V ) ↔ dom { 𝐴 } ≠ ∅ )
3 1 2 bitrdi ( 𝐴𝑉 → ( Rel { 𝐴 } ↔ dom { 𝐴 } ≠ ∅ ) )