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)

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

Proof

Step	Hyp	Ref	Expression
1		dfopif	$$⊢⟨A,B⟩=if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)$$
2		prex	$$⊢\left\{\left\{A\right\},\left\{A,B\right\}\right\}\in \mathrm{V}$$
3		0ex	$$⊢\varnothing \in \mathrm{V}$$
4	2 3	ifex	$$⊢if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)\in \mathrm{V}$$
5	1 4	eqeltri	$$⊢⟨A,B⟩\in \mathrm{V}$$