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Metamath Proof Explorer

Theorem elpredg

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011) (Proof shortened by BJ, 16-Oct-2024)

		Ref	Expression
	Assertion	elpredg	$⊢ (X \in B \land Y \in A) \to (Y \in Pred (R, A, X) \leftrightarrow Y R X)$

Step	Hyp	Ref	Expression
1		elpredgg	$⊢ (X \in B \land Y \in A) \to (Y \in Pred (R, A, X) \leftrightarrow (Y \in A \land Y R X))$
2		ibar	$⊢ Y \in A \to (Y R X \leftrightarrow (Y \in A \land Y R X))$
3	2	bicomd	$⊢ Y \in A \to ((Y \in A \land Y R X) \leftrightarrow Y R X)$
4	3	adantl	$⊢ (X \in B \land Y \in A) \to ((Y \in A \land Y R X) \leftrightarrow Y R X)$
5	1 4	bitrd	$⊢ (X \in B \land Y \in A) \to (Y \in Pred (R, A, X) \leftrightarrow Y R X)$