Theorem ovsng

Description: The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025)

	Ref	Expression
Assertion	ovsng	$$⊢C\in V\to A\left\{⟨⟨A,B⟩,C⟩\right\}B=C$$

Proof

Step	Hyp	Ref	Expression
1		df-ov	$$⊢A\left\{⟨⟨A,B⟩,C⟩\right\}B=\left\{⟨⟨A,B⟩,C⟩\right\}\left(⟨A,B⟩\right)$$
2		opex	$$⊢⟨A,B⟩\in \mathrm{V}$$
3		fvsng	$$⊢\left(⟨A,B⟩\in \mathrm{V}\wedge C\in V\right)\to \left\{⟨⟨A,B⟩,C⟩\right\}\left(⟨A,B⟩\right)=C$$
4	2 3	mpan	$$⊢C\in V\to \left\{⟨⟨A,B⟩,C⟩\right\}\left(⟨A,B⟩\right)=C$$
5	1 4	eqtrid	$$⊢C\in V\to A\left\{⟨⟨A,B⟩,C⟩\right\}B=C$$