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Metamath Proof Explorer

Theorem elioomnf

Description: Membership in an unbounded interval of extended reals. (Contributed by Mario Carneiro, 18-Jun-2014)

Ref Expression
Assertion elioomnf ( 𝐴 ∈ ℝ* → ( 𝐵 ∈ ( -∞ (,) 𝐴 ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 mnfxr ⊢ -∞ ∈ ℝ*
2 elioo2 ⊢ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 ∈ ( -∞ (,) 𝐴 ) ↔ ( 𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴 ) ) )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℝ* → ( 𝐵 ∈ ( -∞ (,) 𝐴 ) ↔ ( 𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴 ) ) )
4 an32 ⊢ ( ( ( 𝐵 ∈ ℝ ∧ -∞ < 𝐵 ) ∧ 𝐵 < 𝐴 ) ↔ ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) ∧ -∞ < 𝐵 ) )
5 df-3an ⊢ ( ( 𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴 ) ↔ ( ( 𝐵 ∈ ℝ ∧ -∞ < 𝐵 ) ∧ 𝐵 < 𝐴 ) )
6 mnflt ⊢ ( 𝐵 ∈ ℝ → -∞ < 𝐵 )
7 6 adantr ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) → -∞ < 𝐵 )
8 7 pm4.71i ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) ↔ ( ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) ∧ -∞ < 𝐵 ) )
9 4 5 8 3bitr4i ⊢ ( ( 𝐵 ∈ ℝ ∧ -∞ < 𝐵 ∧ 𝐵 < 𝐴 ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) )
10 3 9 bitrdi ⊢ ( 𝐴 ∈ ℝ* → ( 𝐵 ∈ ( -∞ (,) 𝐴 ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐵 < 𝐴 ) ) )