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	$$⊢\left(A\in {ℝ}^{*}\wedge A\le B\right)\to \left(B,C\right)\subseteq \left(A,C\right)$$

Proof

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