Theorem relsn

Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013)

	Ref	Expression
Hypothesis	relsn.1	⊢ 𝐴 ∈ V
Assertion	relsn	⊢ ( Rel { 𝐴 } ↔ 𝐴 ∈ ( V × V ) )

Proof

Step	Hyp	Ref	Expression
1		relsn.1	⊢ 𝐴 ∈ V
2		relsng	⊢ ( 𝐴 ∈ V → ( Rel { 𝐴 } ↔ 𝐴 ∈ ( V × V ) ) )
3	1 2	ax-mp	⊢ ( Rel { 𝐴 } ↔ 𝐴 ∈ ( V × V ) )