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 e. _V
Assertion elpredim
|- ( Y e. Pred ( R , A , X ) -> Y R X )

Proof

Step Hyp Ref Expression
1 elpredim.1
 |-  X e. _V
2 elpredimg
 |-  ( ( X e. _V /\ Y e. Pred ( R , A , X ) ) -> Y R X )
3 1 2 mpan
 |-  ( Y e. Pred ( R , A , X ) -> Y R X )