Theorem infxrgelb

Description: The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014) (Revised by AV, 5-Sep-2020)

	Ref	Expression
Assertion	infxrgelb	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ≤ inf ( 𝐴 , ℝ* , < ) ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) )

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 ) )
6	5	notbid	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ¬ inf ( 𝐴 , ℝ* , < ) < 𝐵 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝑥 < 𝐵 ) )
7		ralnex	⊢ ( ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ↔ ¬ ∃ 𝑥 ∈ 𝐴 𝑥 < 𝐵 )
8	6 7	bitr4di	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ¬ inf ( 𝐴 , ℝ* , < ) < 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ) )
9		id	⊢ ( 𝐵 ∈ ℝ* → 𝐵 ∈ ℝ* )
10		infxrcl	⊢ ( 𝐴 ⊆ ℝ* → inf ( 𝐴 , ℝ* , < ) ∈ ℝ* )
11		xrlenlt	⊢ ( ( 𝐵 ∈ ℝ* ∧ inf ( 𝐴 , ℝ* , < ) ∈ ℝ* ) → ( 𝐵 ≤ inf ( 𝐴 , ℝ* , < ) ↔ ¬ inf ( 𝐴 , ℝ* , < ) < 𝐵 ) )
12	9 10 11	syl2anr	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ≤ inf ( 𝐴 , ℝ* , < ) ↔ ¬ inf ( 𝐴 , ℝ* , < ) < 𝐵 ) )
13		simplr	⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ* )
14		simpl	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ⊆ ℝ* )
15	14	sselda	⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ* )
16	13 15	xrlenltd	⊢ ( ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵 ) )
17	16	ralbidva	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ) )
18	8 12 17	3bitr4d	⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 ≤ inf ( 𝐴 , ℝ* , < ) ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) )