Metamath Proof Explorer

Theorem fzolb2

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	fzolb2	$⊢ (M \in ℤ \land N \in ℤ) \to (M \in (M ..^N) \leftrightarrow M < N)$

Proof

Step	Hyp	Ref	Expression
1		fzolb	$⊢ M \in (M ..^N) \leftrightarrow (M \in ℤ \land N \in ℤ \land M < N)$
2		df-3an	$⊢ (M \in ℤ \land N \in ℤ \land M < N) \leftrightarrow ((M \in ℤ \land N \in ℤ) \land M < N)$
3	1 2	bitri	$⊢ M \in (M ..^N) \leftrightarrow ((M \in ℤ \land N \in ℤ) \land M < N)$
4	3	baib	$⊢ (M \in ℤ \land N \in ℤ) \to (M \in (M ..^N) \leftrightarrow M < N)$