Theorem altopth1

Description: Equality of the first members of equal alternate ordered pairs, which holds regardless of the second members' sethood. (Contributed by Scott Fenton, 22-Mar-2012)

	Ref	Expression
Assertion	altopth1	|- ( A e. V -> ( << A , B >> = << C , D >> -> A = C ) )

Proof

Step	Hyp	Ref	Expression
1		altopthsn	|- ( << A , B >> = << C , D >> <-> ( { A } = { C } /\ { B } = { D } ) )
2		sneqrg	|- ( A e. V -> ( { A } = { C } -> A = C ) )
3	2	adantrd	|- ( A e. V -> ( ( { A } = { C } /\ { B } = { D } ) -> A = C ) )
4	1 3	syl5bi	|- ( A e. V -> ( << A , B >> = << C , D >> -> A = C ) )