Theorem opex

Description: An ordered pair of classes is a set. Exercise 7 of TakeutiZaring p. 16. (Contributed by NM, 18-Aug-1993) (Revised by Mario Carneiro, 26-Apr-2015) Avoid ax-nul . (Revised by GG, 6-Mar-2026)

	Ref	Expression
Assertion	opex	$$⊢⟨A,B⟩\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-op	$$⊢⟨A,B⟩=\left\{x|\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\wedge x\in \left\{\left\{A\right\},\left\{A,B\right\}\right\}\right)\right\}$$
2		simp3	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\wedge x\in \left\{\left\{A\right\},\left\{A,B\right\}\right\}\right)\to x\in \left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
3		prex	$$⊢\left\{\left\{A\right\},\left\{A,B\right\}\right\}\in \mathrm{V}$$
4	2 3	abex	$$⊢\left\{x|\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\wedge x\in \left\{\left\{A\right\},\left\{A,B\right\}\right\}\right)\right\}\in \mathrm{V}$$
5	1 4	eqeltri	$$⊢⟨A,B⟩\in \mathrm{V}$$