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Metamath Proof Explorer

Theorem ioossicc

Description: An open interval is a subset of its closure. (Contributed by Paul Chapman, 18-Oct-2007)

		Ref	Expression
	Assertion	ioossicc	$⊢ (A, B) \subseteq [A, B]$

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		xrltle	$⊢ (A \in ℝ^{} \land w \in ℝ^{}) \to (A < w \to A \leq w)$
4		xrltle	$⊢ (w \in ℝ^{} \land B \in ℝ^{}) \to (w < B \to w \leq B)$
5	1 2 3 4	ixxssixx	$⊢ (A, B) \subseteq [A, B]$