iccssioo

Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015)

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

Step	Hyp	Ref	Expression
1		df-ioo	$⊢ (.) = (x \in ℝ^{}, y \in ℝ^{} ⟼ \{z \in ℝ^{*} \| (x < z \land z < y)\})$
2		df-icc	$⊢ [.] = (x \in ℝ^{}, y \in ℝ^{} ⟼ \{z \in ℝ^{*} \| (x \leq z \land z \leq y)\})$
3		xrltletr	$⊢ (A \in ℝ^{} \land C \in ℝ^{} \land w \in ℝ^{*}) \to ((A < C \land C \leq w) \to A < w)$
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	$⊢ ((A \in ℝ^{} \land B \in ℝ^{}) \land (A < C \land D < B)) \to [C, D] \subseteq (A, B)$

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