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Metamath Proof Explorer

Theorem fimaxre2

Description: A nonempty finite set of real numbers has an upper bound. (Contributed by Jeff Madsen, 27-May-2011) (Revised by Mario Carneiro, 13-Feb-2014)

Ref Expression
Assertion fimaxre2 ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )

Proof

Step Hyp Ref Expression
1 0re ⊢ 0 ∈ ℝ
2 rzal ⊢ ( 𝐴 = ∅ → ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 0 )
3 brralrspcev ⊢ ( ( 0 ∈ ℝ ∧ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 0 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
4 1 2 3 sylancr ⊢ ( 𝐴 = ∅ → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
5 4 a1i ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ( 𝐴 = ∅ → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) )
6 fimaxre ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
7 6 3expia ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) )
8 ssrexv ⊢ ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) )
9 8 adantr ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) )
10 7 9 syld ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) )
11 5 10 pm2.61dne ⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )