Theorem dfop

Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998) (Avoid depending on this detail.)

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

Proof

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