Theorem elpred

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011) (Proof shortened by BJ, 16-Oct-2024)

	Ref	Expression
Hypothesis	elpred.1	$$⊢Y\in \mathrm{V}$$
Assertion	elpred	$$⊢X\in D\to \left(Y\in Pred\left(R,A,X\right)\leftrightarrow \left(Y\in A\wedge YRX\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elpred.1	$$⊢Y\in \mathrm{V}$$
2		elpredgg	$$⊢\left(X\in D\wedge Y\in \mathrm{V}\right)\to \left(Y\in Pred\left(R,A,X\right)\leftrightarrow \left(Y\in A\wedge YRX\right)\right)$$
3	1 2	mpan2	$$⊢X\in D\to \left(Y\in Pred\left(R,A,X\right)\leftrightarrow \left(Y\in A\wedge YRX\right)\right)$$