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	$$⊢\neg \varnothing \in \left(A/R\right)\leftrightarrow \mathrm{dom}\left({R\upharpoonright }_A\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		n0elqs	$$⊢\neg \varnothing \in \left(A/R\right)\leftrightarrow A\subseteq \mathrm{dom}R$$
2		ssdmres	$$⊢A\subseteq \mathrm{dom}R\leftrightarrow \mathrm{dom}\left({R\upharpoonright }_A\right)=A$$
3	1 2	bitri	$$⊢\neg \varnothing \in \left(A/R\right)\leftrightarrow \mathrm{dom}\left({R\upharpoonright }_A\right)=A$$