Theorem opideq

Description: Equality conditions for ordered pairs <. A , A >. and <. B , B >. . (Contributed by Peter Mazsa, 22-Jul-2019) (Revised by Thierry Arnoux, 16-Feb-2022)

	Ref	Expression
Assertion	opideq	|- ( A e. V -> ( <. A , A >. = <. B , B >. <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		opthg	|- ( ( A e. V /\ A e. V ) -> ( <. A , A >. = <. B , B >. <-> ( A = B /\ A = B ) ) )
2	1	anidms	|- ( A e. V -> ( <. A , A >. = <. B , B >. <-> ( A = B /\ A = B ) ) )
3		anidm	|- ( ( A = B /\ A = B ) <-> A = B )
4	2 3	syl6bb	|- ( A e. V -> ( <. A , A >. = <. B , B >. <-> A = B ) )