Metamath Proof Explorer


Theorem supxrub

Description: A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006)

Ref Expression
Assertion supxrub ( ( 𝐴 ⊆ ℝ*𝐵𝐴 ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) )

Proof

Step Hyp Ref Expression
1 ssel2 ( ( 𝐴 ⊆ ℝ*𝐵𝐴 ) → 𝐵 ∈ ℝ* )
2 supxrcl ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )
3 2 adantr ( ( 𝐴 ⊆ ℝ*𝐵𝐴 ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )
4 xrltso < Or ℝ*
5 4 a1i ( 𝐴 ⊆ ℝ* → < Or ℝ* )
6 xrsupss ( 𝐴 ⊆ ℝ* → ∃ 𝑥 ∈ ℝ* ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) ) )
7 5 6 supub ( 𝐴 ⊆ ℝ* → ( 𝐵𝐴 → ¬ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) )
8 7 imp ( ( 𝐴 ⊆ ℝ*𝐵𝐴 ) → ¬ sup ( 𝐴 , ℝ* , < ) < 𝐵 )
9 1 3 8 xrnltled ( ( 𝐴 ⊆ ℝ*𝐵𝐴 ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) )