Theorem fvsn

Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994) (Proof shortened by BJ, 25-Feb-2023)

	Ref	Expression
Hypotheses	fvsn.1	$$⊢A\in \mathrm{V}$$
	fvsn.2	$$⊢B\in \mathrm{V}$$
Assertion	fvsn	$$⊢\left\{⟨A,B⟩\right\}\left(A\right)=B$$

Proof

Step	Hyp	Ref	Expression
1		fvsn.1	$$⊢A\in \mathrm{V}$$
2		fvsn.2	$$⊢B\in \mathrm{V}$$
3		fvsng	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left\{⟨A,B⟩\right\}\left(A\right)=B$$
4	1 2 3	mp2an	$$⊢\left\{⟨A,B⟩\right\}\left(A\right)=B$$