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