iocssioo

Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017)

		Ref	Expression
	Assertion	iocssioo	$⊢ ((A \in ℝ^{} \land B \in ℝ^{}) \land (A \leq C \land D < B)) \to (C, D] \subseteq (A, B)$

Step	Hyp	Ref	Expression
1		df-ioo	$⊢ (.) = (a \in ℝ^{}, b \in ℝ^{} ⟼ \{x \in ℝ^{*} \| (a < x \land x < b)\})$
2		df-ioc	$⊢ (.] = (a \in ℝ^{}, b \in ℝ^{} ⟼ \{x \in ℝ^{*} \| (a < x \land x \leq b)\})$
3		xrlelttr	$⊢ (A \in ℝ^{} \land C \in ℝ^{} \land w \in ℝ^{*}) \to ((A \leq C \land C < w) \to A < w)$
4		xrlelttr	$⊢ (w \in ℝ^{} \land D \in ℝ^{} \land B \in ℝ^{*}) \to ((w \leq D \land D < B) \to w < B)$
5	1 2 3 4	ixxss12	$⊢ ((A \in ℝ^{} \land B \in ℝ^{}) \land (A \leq C \land D < B)) \to (C, D] \subseteq (A, B)$

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