This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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	$⊢ (X \in V \land Y \in Pred (R, A, X)) \to Y R X$

Proof

Step	Hyp	Ref	Expression
1		elpredgg	$⊢ (X \in V \land Y \in Pred (R, A, X)) \to (Y \in Pred (R, A, X) \leftrightarrow (Y \in A \land Y R X))$
2		simpr	$⊢ (Y \in A \land Y R X) \to Y R X$
3	1 2	biimtrdi	$⊢ (X \in V \land Y \in Pred (R, A, X)) \to (Y \in Pred (R, A, X) \to Y R X)$
4	3	syldbl2	$⊢ (X \in V \land Y \in Pred (R, A, X)) \to Y R X$