Theorem fzolb

Description: The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with M < N . This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate M e. ( ZZ>=N ) . (Contributed by Mario Carneiro, 29-Sep-2015)

	Ref	Expression
Assertion	fzolb	|- ( M e. ( M ..^ N ) <-> ( M e. ZZ /\ N e. ZZ /\ M < N ) )

Proof

Step	Hyp	Ref	Expression
1		elfzo2	|- ( M e. ( M ..^ N ) <-> ( M e. ( ZZ>= ` M ) /\ N e. ZZ /\ M < N ) )
2		eluzel2	|- ( M e. ( ZZ>= ` M ) -> M e. ZZ )
3		uzid	|- ( M e. ZZ -> M e. ( ZZ>= ` M ) )
4	2 3	impbii	|- ( M e. ( ZZ>= ` M ) <-> M e. ZZ )
5	4	3anbi1i	|- ( ( M e. ( ZZ>= ` M ) /\ N e. ZZ /\ M < N ) <-> ( M e. ZZ /\ N e. ZZ /\ M < N ) )
6	1 5	bitri	|- ( M e. ( M ..^ N ) <-> ( M e. ZZ /\ N e. ZZ /\ M < N ) )