Theorem fzssfzo

Description: Condition for an integer interval to be a subset of a half-open integer interval. (Contributed by Thierry Arnoux, 8-Oct-2018)

	Ref	Expression
Assertion	fzssfzo	|- ( K e. ( M ..^ N ) -> ( M ... K ) C_ ( M ..^ N ) )

Proof

Step	Hyp	Ref	Expression
1		elfzoel2	|- ( K e. ( M ..^ N ) -> N e. ZZ )
2		fzoval	|- ( N e. ZZ -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
3	1 2	syl	|- ( K e. ( M ..^ N ) -> ( M ..^ N ) = ( M ... ( N - 1 ) ) )
4	3	eleq2d	|- ( K e. ( M ..^ N ) -> ( K e. ( M ..^ N ) <-> K e. ( M ... ( N - 1 ) ) ) )
5	4	ibi	|- ( K e. ( M ..^ N ) -> K e. ( M ... ( N - 1 ) ) )
6		elfzuz3	|- ( K e. ( M ... ( N - 1 ) ) -> ( N - 1 ) e. ( ZZ>= ` K ) )
7		fzss2	|- ( ( N - 1 ) e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... ( N - 1 ) ) )
8	5 6 7	3syl	|- ( K e. ( M ..^ N ) -> ( M ... K ) C_ ( M ... ( N - 1 ) ) )
9	8 3	sseqtrrd	|- ( K e. ( M ..^ N ) -> ( M ... K ) C_ ( M ..^ N ) )