Theorem infxrglb

Description: The infimum of a set of extended reals is less than an extended real if and only if the set contains a smaller number. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Assertion	infxrglb	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( inf ( 𝐴 , ℝ* , < ) < 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑥 < 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		xrltso	⊢ < Or ℝ*
2	1	a1i	⊢ ( 𝐴 ⊆ ℝ* → < Or ℝ* )
3		xrinfmss	⊢ ( 𝐴 ⊆ ℝ* → ∃ 𝑧 ∈ ℝ* ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑧 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑧 < 𝑦 → ∃ 𝑥 ∈ 𝐴 𝑥 < 𝑦 ) ) )
4		id	⊢ ( 𝐴 ⊆ ℝ* → 𝐴 ⊆ ℝ* )
5	2 3 4	infglbb	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( inf ( 𝐴 , ℝ* , < ) < 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑥 < 𝐵 ) )