Metamath Proof Explorer


Theorem elfzle1

Description: A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfzle1
|- ( K e. ( M ... N ) -> M <_ K )

Proof

Step Hyp Ref Expression
1 elfzuz
 |-  ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
2 eluzle
 |-  ( K e. ( ZZ>= ` M ) -> M <_ K )
3 1 2 syl
 |-  ( K e. ( M ... N ) -> M <_ K )