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Metamath Proof Explorer

Theorem elpredim

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012) (Proof shortened by BJ, 16-Oct-2024)

		Ref	Expression
	Hypothesis	elpredim.1	$⊢ X \in V$
	Assertion	elpredim	$⊢ Y \in Pred (R, A, X) \to Y R X$

Proof

Step	Hyp	Ref	Expression
1		elpredim.1	$⊢ X \in V$
2		elpredimg	$⊢ (X \in V \land Y \in Pred (R, A, X)) \to Y R X$
3	1 2	mpan	$⊢ Y \in Pred (R, A, X) \to Y R X$