Theorem supiccub

Description: The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019)

	Ref	Expression
Hypotheses	supicc.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	supicc.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
	supicc.3	⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐵 [,] 𝐶 ) )
	supicc.4	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
	supiccub.1	⊢ ( 𝜑 → 𝐷 ∈ 𝐴 )
Assertion	supiccub	⊢ ( 𝜑 → 𝐷 ≤ sup ( 𝐴 , ℝ , < ) )

Proof

Step	Hyp	Ref	Expression
1		supicc.1	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
2		supicc.2	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
3		supicc.3	⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐵 [,] 𝐶 ) )
4		supicc.4	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
5		supiccub.1	⊢ ( 𝜑 → 𝐷 ∈ 𝐴 )
6		iccssre	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 [,] 𝐶 ) ⊆ ℝ )
7	1 2 6	syl2anc	⊢ ( 𝜑 → ( 𝐵 [,] 𝐶 ) ⊆ ℝ )
8	3 7	sstrd	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
9	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
10	9	rexrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ* )
11	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℝ )
12	11	rexrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℝ* )
13	3	sselda	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ( 𝐵 [,] 𝐶 ) )
14		iccleub	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) → 𝑥 ≤ 𝐶 )
15	10 12 13 14	syl3anc	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≤ 𝐶 )
16	15	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝑥 ≤ 𝐶 )
17		brralrspcev	⊢ ( ( 𝐶 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝑥 ≤ 𝐶 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑦 )
18	2 16 17	syl2anc	⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑥 ≤ 𝑦 )
19	8 4 18 5	suprubd	⊢ ( 𝜑 → 𝐷 ≤ sup ( 𝐴 , ℝ , < ) )