Theorem suprleub

Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005) (Revised by Mario Carneiro, 6-Sep-2014)

	Ref	Expression
Assertion	suprleub	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		suprnub	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < sup ( 𝐴 , ℝ , < ) ↔ ∀ 𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ) )
2		suprcl	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → sup ( 𝐴 , ℝ , < ) ∈ ℝ )
3		lenlt	⊢ ( ( sup ( 𝐴 , ℝ , < ) ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup ( 𝐴 , ℝ , < ) ) )
4	2 3	sylan	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ¬ 𝐵 < sup ( 𝐴 , ℝ , < ) ) )
5		simpl1	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → 𝐴 ⊆ ℝ )
6	5	sselda	⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ ℝ )
7		simplr	⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) ∧ 𝑤 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
8	6 7	lenltd	⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤 ) )
9	8	ralbidva	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( ∀ 𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀ 𝑤 ∈ 𝐴 ¬ 𝐵 < 𝑤 ) )
10	1 4 9	3bitr4d	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ∀ 𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ) )
11		breq1	⊢ ( 𝑤 = 𝑧 → ( 𝑤 ≤ 𝐵 ↔ 𝑧 ≤ 𝐵 ) )
12	11	cbvralvw	⊢ ( ∀ 𝑤 ∈ 𝐴 𝑤 ≤ 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 )
13	10 12	bitrdi	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( sup ( 𝐴 , ℝ , < ) ≤ 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ≤ 𝐵 ) )