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Metamath Proof Explorer

Theorem iooss1

Description: Subset relationship for open intervals of extended reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 20-Feb-2015)

		Ref	Expression
	Assertion	iooss1	$⊢ (A \in ℝ^{*} \land A \leq B) \to (B, C) \subseteq (A, C)$

Step	Hyp	Ref	Expression
1		df-ioo	$⊢ (.) = (x \in ℝ^{}, y \in ℝ^{} ⟼ \{z \in ℝ^{*} \| (x < z \land z < y)\})$
2		xrlelttr	$⊢ (A \in ℝ^{} \land B \in ℝ^{} \land w \in ℝ^{*}) \to ((A \leq B \land B < w) \to A < w)$
3	1 1 2	ixxss1	$⊢ (A \in ℝ^{*} \land A \leq B) \to (B, C) \subseteq (A, C)$