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Metamath Proof Explorer

Theorem elioo5

Description: Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008)

Ref Expression
Assertion elioo5 ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 elioo1 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
2 1 3adant3 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
3 3anass ⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
4 3 baibr ⊢ ( 𝐶 ∈ ℝ* → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
5 4 3ad2ant3 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )
6 2 5 bitr4d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) )