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 ) ) )