Theorem elfzr

Description: A member of a finite interval of integers is either a member of the corresponding half-open integer range or the upper bound of the interval. (Contributed by AV, 5-Feb-2021)

	Ref	Expression
Assertion	elfzr	|- ( K e. ( M ... N ) -> ( K e. ( M ..^ N ) \/ K = N ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuz2	|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` M ) )
2		fzisfzounsn	|- ( N e. ( ZZ>= ` M ) -> ( M ... N ) = ( ( M ..^ N ) u. { N } ) )
3	2	eleq2d	|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) <-> K e. ( ( M ..^ N ) u. { N } ) ) )
4		elun	|- ( K e. ( ( M ..^ N ) u. { N } ) <-> ( K e. ( M ..^ N ) \/ K e. { N } ) )
5		elsni	|- ( K e. { N } -> K = N )
6	5	orim2i	|- ( ( K e. ( M ..^ N ) \/ K e. { N } ) -> ( K e. ( M ..^ N ) \/ K = N ) )
7	4 6	sylbi	|- ( K e. ( ( M ..^ N ) u. { N } ) -> ( K e. ( M ..^ N ) \/ K = N ) )
8	3 7	biimtrdi	|- ( N e. ( ZZ>= ` M ) -> ( K e. ( M ... N ) -> ( K e. ( M ..^ N ) \/ K = N ) ) )
9	1 8	mpcom	|- ( K e. ( M ... N ) -> ( K e. ( M ..^ N ) \/ K = N ) )