Theorem elpredimg

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 13-Apr-2011) (Revised by NM, 5-Apr-2016) (Proof shortened by BJ, 16-Oct-2024)

	Ref	Expression
Assertion	elpredimg	|- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> Y R X )

Proof

Step	Hyp	Ref	Expression
1		elpredgg	|- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> ( Y e. Pred ( R , A , X ) <-> ( Y e. A /\ Y R X ) ) )
2		simpr	|- ( ( Y e. A /\ Y R X ) -> Y R X )
3	1 2	biimtrdi	|- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> ( Y e. Pred ( R , A , X ) -> Y R X ) )
4	3	syldbl2	|- ( ( X e. V /\ Y e. Pred ( R , A , X ) ) -> Y R X )