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Metamath Proof Explorer

Theorem ioossioo

Description: Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017)

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

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