Theorem relsnopg

Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998) (Revised by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	relsnopg	$$⊢\left(A\in V\wedge B\in W\right)\to \mathrm{Rel}\left\{⟨A,B⟩\right\}$$

Proof

Step	Hyp	Ref	Expression
1		opelvvg	$$⊢\left(A\in V\wedge B\in W\right)\to ⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$
2		opex	$$⊢⟨A,B⟩\in \mathrm{V}$$
3		relsng	$$⊢⟨A,B⟩\in \mathrm{V}\to \left(\mathrm{Rel}\left\{⟨A,B⟩\right\}\leftrightarrow ⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)\right)$$
4	2 3	mp1i	$$⊢\left(A\in V\wedge B\in W\right)\to \left(\mathrm{Rel}\left\{⟨A,B⟩\right\}\leftrightarrow ⟨A,B⟩\in \left(\mathrm{V}\times \mathrm{V}\right)\right)$$
5	1 4	mpbird	$$⊢\left(A\in V\wedge B\in W\right)\to \mathrm{Rel}\left\{⟨A,B⟩\right\}$$