Theorem n0elqs2

Description: Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 25-Jul-2021)

	Ref	Expression
Assertion	n0elqs2	⊢ ( ¬ ∅ ∈ ( 𝐴 / 𝑅 ) ↔ dom ( 𝑅 ↾ 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		n0elqs	⊢ ( ¬ ∅ ∈ ( 𝐴 / 𝑅 ) ↔ 𝐴 ⊆ dom 𝑅 )
2		ssdmres	⊢ ( 𝐴 ⊆ dom 𝑅 ↔ dom ( 𝑅 ↾ 𝐴 ) = 𝐴 )
3	1 2	bitri	⊢ ( ¬ ∅ ∈ ( 𝐴 / 𝑅 ) ↔ dom ( 𝑅 ↾ 𝐴 ) = 𝐴 )