Theorem lbinfcl

Description: If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005) (Revised by AV, 4-Sep-2020)

	Ref	Expression
Assertion	lbinfcl	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → inf ( 𝑆 , ℝ , < ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		lbinf	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → inf ( 𝑆 , ℝ , < ) = ( ℩ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) )
2		lbcl	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → ( ℩ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) ∈ 𝑆 )
3	1 2	eqeltrd	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → inf ( 𝑆 , ℝ , < ) ∈ 𝑆 )