# Metamath Proof Explorer

## Theorem supxrleub

Description: The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006) (Revised by Mario Carneiro, 6-Sep-2014)

Ref Expression
Assertion supxrleub ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( sup ( 𝐴 , ℝ* , < ) ≤ 𝐵 ↔ ∀ 𝑥𝐴 𝑥𝐵 ) )

### Proof

Step Hyp Ref Expression
1 supxrlub ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( 𝐵 < sup ( 𝐴 , ℝ* , < ) ↔ ∃ 𝑥𝐴 𝐵 < 𝑥 ) )
2 1 notbid ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( ¬ 𝐵 < sup ( 𝐴 , ℝ* , < ) ↔ ¬ ∃ 𝑥𝐴 𝐵 < 𝑥 ) )
3 ralnex ( ∀ 𝑥𝐴 ¬ 𝐵 < 𝑥 ↔ ¬ ∃ 𝑥𝐴 𝐵 < 𝑥 )
4 2 3 syl6bbr ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( ¬ 𝐵 < sup ( 𝐴 , ℝ* , < ) ↔ ∀ 𝑥𝐴 ¬ 𝐵 < 𝑥 ) )
5 supxrcl ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* )
6 xrlenlt ( ( sup ( 𝐴 , ℝ* , < ) ∈ ℝ*𝐵 ∈ ℝ* ) → ( sup ( 𝐴 , ℝ* , < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup ( 𝐴 , ℝ* , < ) ) )
7 5 6 sylan ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( sup ( 𝐴 , ℝ* , < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup ( 𝐴 , ℝ* , < ) ) )
8 simpl ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → 𝐴 ⊆ ℝ* )
9 8 sselda ( ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) ∧ 𝑥𝐴 ) → 𝑥 ∈ ℝ* )
10 simplr ( ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) ∧ 𝑥𝐴 ) → 𝐵 ∈ ℝ* )
11 9 10 xrlenltd ( ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) ∧ 𝑥𝐴 ) → ( 𝑥𝐵 ↔ ¬ 𝐵 < 𝑥 ) )
12 11 ralbidva ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( ∀ 𝑥𝐴 𝑥𝐵 ↔ ∀ 𝑥𝐴 ¬ 𝐵 < 𝑥 ) )
13 4 7 12 3bitr4d ( ( 𝐴 ⊆ ℝ*𝐵 ∈ ℝ* ) → ( sup ( 𝐴 , ℝ* , < ) ≤ 𝐵 ↔ ∀ 𝑥𝐴 𝑥𝐵 ) )