Theorem brsnop

Description: Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023)

	Ref	Expression
Assertion	brsnop	$$⊢\left(A\in V\wedge B\in W\right)\to \left(X\left\{⟨A,B⟩\right\}Y\leftrightarrow \left(X=A\wedge Y=B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-br	$$⊢X\left\{⟨A,B⟩\right\}Y\leftrightarrow ⟨X,Y⟩\in \left\{⟨A,B⟩\right\}$$
2		opex	$$⊢⟨X,Y⟩\in \mathrm{V}$$
3	2	elsn	$$⊢⟨X,Y⟩\in \left\{⟨A,B⟩\right\}\leftrightarrow ⟨X,Y⟩=⟨A,B⟩$$
4		opthg2	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨X,Y⟩=⟨A,B⟩\leftrightarrow \left(X=A\wedge Y=B\right)\right)$$
5	3 4	syl5bb	$$⊢\left(A\in V\wedge B\in W\right)\to \left(⟨X,Y⟩\in \left\{⟨A,B⟩\right\}\leftrightarrow \left(X=A\wedge Y=B\right)\right)$$
6	1 5	syl5bb	$$⊢\left(A\in V\wedge B\in W\right)\to \left(X\left\{⟨A,B⟩\right\}Y\leftrightarrow \left(X=A\wedge Y=B\right)\right)$$