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 e. V -> <. A , A >. = { { A } } )

Proof

Step	Hyp	Ref	Expression
1		dfopg	|- ( ( A e. V /\ A e. V ) -> <. A , A >. = { { A } , { A , A } } )
2	1	anidms	|- ( A e. V -> <. A , A >. = { { A } , { A , A } } )
3		dfsn2	|- { A } = { A , A }
4	3	eqcomi	|- { A , A } = { A }
5	4	preq2i	|- { { A } , { A , A } } = { { A } , { A } }
6		dfsn2	|- { { A } } = { { A } , { A } }
7	5 6	eqtr4i	|- { { A } , { A , A } } = { { A } }
8	2 7	eqtrdi	|- ( A e. V -> <. A , A >. = { { A } } )