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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* ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) )

Proof

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