Metamath Proof Explorer


Theorem eltrpred

Description: A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not meant to be used directly; use trpredpred and trpredmintr instead. (Contributed by Scott Fenton, 28-Apr-2012)

Ref Expression
Assertion eltrpred YTrPredRAXiωYrecaVyaPredRAyPredRAXωi

Proof

Step Hyp Ref Expression
1 dftrpred2 TrPredRAX=iωrecaVyaPredRAyPredRAXωi
2 1 eleq2i YTrPredRAXYiωrecaVyaPredRAyPredRAXωi
3 eliun YiωrecaVyaPredRAyPredRAXωiiωYrecaVyaPredRAyPredRAXωi
4 2 3 bitri YTrPredRAXiωYrecaVyaPredRAyPredRAXωi