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	$$⊢X\in \mathrm{V}$$
Assertion	elpredim	$$⊢Y\in Pred\left(R,A,X\right)\to YRX$$

Proof

Step	Hyp	Ref	Expression
1		elpredim.1	$$⊢X\in \mathrm{V}$$
2		elpredimg	$$⊢\left(X\in \mathrm{V}\wedge Y\in Pred\left(R,A,X\right)\right)\to YRX$$
3	1 2	mpan	$$⊢Y\in Pred\left(R,A,X\right)\to YRX$$