Theorem eliood

Description: Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	eliood.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	eliood.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	eliood.3	$$⊢\phi \to C\in ℝ$$
	eliood.4	$$⊢\phi \to A<C$$
	eliood.5	$$⊢\phi \to C<B$$
Assertion	eliood	$$⊢\phi \to C\in \left(A,B\right)$$

Proof

Step	Hyp	Ref	Expression
1		eliood.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		eliood.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		eliood.3	$$⊢\phi \to C\in ℝ$$
4		eliood.4	$$⊢\phi \to A<C$$
5		eliood.5	$$⊢\phi \to C<B$$
6		elioo2	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to \left(C\in \left(A,B\right)\leftrightarrow \left(C\in ℝ\wedge A<C\wedge C<B\right)\right)$$
7	1 2 6	syl2anc	$$⊢\phi \to \left(C\in \left(A,B\right)\leftrightarrow \left(C\in ℝ\wedge A<C\wedge C<B\right)\right)$$
8	3 4 5 7	mpbir3and	$$⊢\phi \to C\in \left(A,B\right)$$