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	$$⊢\left(X\in V\wedge Y\in Pred\left(R,A,X\right)\right)\to YRX$$

Proof

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