Theorem opidg

Description: The ordered pair <. A , A >. in Kuratowski's representation. Closed form of opid . (Contributed by Peter Mazsa, 22-Jul-2019) (Avoid depending on this detail.)

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

Proof

Step	Hyp	Ref	Expression
1		dfopg	$$⊢\left(A\in V\wedge A\in V\right)\to ⟨A,A⟩=\left\{\left\{A\right\},\left\{A,A\right\}\right\}$$
2	1	anidms	$$⊢A\in V\to ⟨A,A⟩=\left\{\left\{A\right\},\left\{A,A\right\}\right\}$$
3		dfsn2	$$⊢\left\{A\right\}=\left\{A,A\right\}$$
4	3	eqcomi	$$⊢\left\{A,A\right\}=\left\{A\right\}$$
5	4	preq2i	$$⊢\left\{\left\{A\right\},\left\{A,A\right\}\right\}=\left\{\left\{A\right\},\left\{A\right\}\right\}$$
6		dfsn2	$$⊢\left\{\left\{A\right\}\right\}=\left\{\left\{A\right\},\left\{A\right\}\right\}$$
7	5 6	eqtr4i	$$⊢\left\{\left\{A\right\},\left\{A,A\right\}\right\}=\left\{\left\{A\right\}\right\}$$
8	2 7	eqtrdi	$$⊢A\in V\to ⟨A,A⟩=\left\{\left\{A\right\}\right\}$$