Theorem lbcl

Description: If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005) (Revised by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Assertion	lbcl	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → ( ℩ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		lbreu	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → ∃! 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 )
2		riotacl	⊢ ( ∃! 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → ( ℩ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) ∈ 𝑆 )
3	1 2	syl	⊢ ( ( 𝑆 ⊆ ℝ ∧ ∃ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) → ( ℩ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ) ∈ 𝑆 )