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Metamath Proof Explorer

Theorem elicopnf

Description: Membership in a closed unbounded interval of reals. (Contributed by Mario Carneiro, 16-Sep-2014)

Ref Expression
Assertion elicopnf ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ( 𝐴 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 pnfxr ⊢ +∞ ∈ ℝ*
2 elico2 ⊢ ( ( 𝐴 ∈ ℝ ∧ +∞ ∈ ℝ* ) → ( 𝐵 ∈ ( 𝐴 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) ) )
3 1 2 mpan2 ⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ( 𝐴 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) ) )
4 ltpnf ⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ )
5 4 adantr ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐵 < +∞ )
6 5 pm4.71i ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ↔ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐵 < +∞ ) )
7 df-3an ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) ↔ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐵 < +∞ ) )
8 6 7 bitr4i ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 < +∞ ) )
9 3 8 bitr4di ⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ( 𝐴 [,) +∞ ) ↔ ( 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) ) )