Theorem rnsnn0

Description: The range of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008)

	Ref	Expression
Assertion	rnsnn0	⊢ ( 𝐴 ∈ ( V × V ) ↔ ran { 𝐴 } ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		dmsnn0	⊢ ( 𝐴 ∈ ( V × V ) ↔ dom { 𝐴 } ≠ ∅ )
2		dm0rn0	⊢ ( dom { 𝐴 } = ∅ ↔ ran { 𝐴 } = ∅ )
3	2	necon3bii	⊢ ( dom { 𝐴 } ≠ ∅ ↔ ran { 𝐴 } ≠ ∅ )
4	1 3	bitri	⊢ ( 𝐴 ∈ ( V × V ) ↔ ran { 𝐴 } ≠ ∅ )