This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem elioopnf

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

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

Proof

Step Hyp Ref Expression
1 pnfxr ⊢ +∞ ∈ ℝ*
2 elioo2 ⊢ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝐵 ∈ ( 𝐴 (,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞ ) ) )
3 1 2 mpan2 ⊢ ( 𝐴 ∈ ℝ* → ( 𝐵 ∈ ( 𝐴 (,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞ ) ) )
4 df-3an ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞ ) ↔ ( ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) ∧ 𝐵 < +∞ ) )
5 ltpnf ⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ )
6 5 adantr ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → 𝐵 < +∞ )
7 6 pm4.71i ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) ↔ ( ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) ∧ 𝐵 < +∞ ) )
8 4 7 bitr4i ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ∧ 𝐵 < +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) )
9 3 8 bitrdi ⊢ ( 𝐴 ∈ ℝ* → ( 𝐵 ∈ ( 𝐴 (,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) ) )