Metamath Proof Explorer


Theorem fri

Description: A nonempty subset of an R -well-founded class has an R -minimal element (inference form). (Contributed by BJ, 16-Nov-2024) (Proof shortened by BJ, 19-Nov-2024)

Ref Expression
Assertion fri
|- ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )

Proof

Step Hyp Ref Expression
1 simplr
 |-  ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> R Fr A )
2 simprl
 |-  ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> B C_ A )
3 simpll
 |-  ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> B e. C )
4 simprr
 |-  ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> B =/= (/) )
5 1 2 3 4 frd
 |-  ( ( ( B e. C /\ R Fr A ) /\ ( B C_ A /\ B =/= (/) ) ) -> E. x e. B A. y e. B -. y R x )