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Metamath Proof Explorer

Theorem iccssre

Description: A closed real interval is a set of reals. (Contributed by FL, 6-Jun-2007) (Proof shortened by Paul Chapman, 21-Jan-2008)

		Ref	Expression
	Assertion	iccssre	$⊢ (A \in ℝ \land B \in ℝ) \to [A, B] \subseteq ℝ$

Step	Hyp	Ref	Expression
1		elicc2	$⊢ (A \in ℝ \land B \in ℝ) \to (x \in [A, B] \leftrightarrow (x \in ℝ \land A \leq x \land x \leq B))$
2	1	biimp3a	$⊢ (A \in ℝ \land B \in ℝ \land x \in [A, B]) \to (x \in ℝ \land A \leq x \land x \leq B)$
3	2	simp1d	$⊢ (A \in ℝ \land B \in ℝ \land x \in [A, B]) \to x \in ℝ$
4	3	3expia	$⊢ (A \in ℝ \land B \in ℝ) \to (x \in [A, B] \to x \in ℝ)$
5	4	ssrdv	$⊢ (A \in ℝ \land B \in ℝ) \to [A, B] \subseteq ℝ$