Theorem opi2

Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 26-Apr-2015) (Avoid depending on this detail.)

	Ref	Expression
Hypotheses	opi1.1	$$⊢A\in \mathrm{V}$$
	opi1.2	$$⊢B\in \mathrm{V}$$
Assertion	opi2	$$⊢\left\{A,B\right\}\in ⟨A,B⟩$$

Proof

Step	Hyp	Ref	Expression
1		opi1.1	$$⊢A\in \mathrm{V}$$
2		opi1.2	$$⊢B\in \mathrm{V}$$
3		prex	$$⊢\left\{A,B\right\}\in \mathrm{V}$$
4	3	prid2	$$⊢\left\{A,B\right\}\in \left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
5	1 2	dfop	$$⊢⟨A,B⟩=\left\{\left\{A\right\},\left\{A,B\right\}\right\}$$
6	4 5	eleqtrri	$$⊢\left\{A,B\right\}\in ⟨A,B⟩$$