Theorem elrel

Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006)

	Ref	Expression
Assertion	elrel	$$⊢\left(\mathrm{Rel}R\wedge A\in R\right)\to \exists x\exists yA=⟨x,y⟩$$

Proof

Step	Hyp	Ref	Expression
1		df-rel	$$⊢\mathrm{Rel}R\leftrightarrow R\subseteq \mathrm{V}\times \mathrm{V}$$
2	1	biimpi	$$⊢\mathrm{Rel}R\to R\subseteq \mathrm{V}\times \mathrm{V}$$
3	2	sselda	$$⊢\left(\mathrm{Rel}R\wedge A\in R\right)\to A\in \left(\mathrm{V}\times \mathrm{V}\right)$$
4		elvv	$$⊢A\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \exists x\exists yA=⟨x,y⟩$$
5	3 4	sylib	$$⊢\left(\mathrm{Rel}R\wedge A\in R\right)\to \exists x\exists yA=⟨x,y⟩$$