Theorem relrn0

Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004)

	Ref	Expression
Assertion	relrn0	⊢ ( Rel 𝐴 → ( 𝐴 = ∅ ↔ ran 𝐴 = ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		reldm0	⊢ ( Rel 𝐴 → ( 𝐴 = ∅ ↔ dom 𝐴 = ∅ ) )
2		dm0rn0	⊢ ( dom 𝐴 = ∅ ↔ ran 𝐴 = ∅ )
3	1 2	bitrdi	⊢ ( Rel 𝐴 → ( 𝐴 = ∅ ↔ ran 𝐴 = ∅ ) )