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Metamath Proof Explorer

Theorem elioo3g

Description: Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show A e. RR* and B e. RR* . (Contributed by NM, 24-Dec-2006) (Revised by Mario Carneiro, 3-Nov-2013)

		Ref	Expression
	Assertion	elioo3g	⊢ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
2	1	elixx3g	⊢ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )