Metamath Proof Explorer


Theorem xrge0infssd

Description: Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019) (Revised by AV, 4-Oct-2020)

Ref Expression
Hypotheses xrge0infssd.1 ( 𝜑𝐶𝐵 )
xrge0infssd.2 ( 𝜑𝐵 ⊆ ( 0 [,] +∞ ) )
Assertion xrge0infssd ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ≤ inf ( 𝐶 , ( 0 [,] +∞ ) , < ) )

Proof

Step Hyp Ref Expression
1 xrge0infssd.1 ( 𝜑𝐶𝐵 )
2 xrge0infssd.2 ( 𝜑𝐵 ⊆ ( 0 [,] +∞ ) )
3 iccssxr ( 0 [,] +∞ ) ⊆ ℝ*
4 xrltso < Or ℝ*
5 soss ( ( 0 [,] +∞ ) ⊆ ℝ* → ( < Or ℝ* → < Or ( 0 [,] +∞ ) ) )
6 3 4 5 mp2 < Or ( 0 [,] +∞ )
7 6 a1i ( 𝜑 → < Or ( 0 [,] +∞ ) )
8 xrge0infss ( 𝐵 ⊆ ( 0 [,] +∞ ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦𝐵 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧𝐵 𝑧 < 𝑦 ) ) )
9 2 8 syl ( 𝜑 → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦𝐵 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧𝐵 𝑧 < 𝑦 ) ) )
10 7 9 infcl ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ∈ ( 0 [,] +∞ ) )
11 3 10 sselid ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ∈ ℝ* )
12 1 2 sstrd ( 𝜑𝐶 ⊆ ( 0 [,] +∞ ) )
13 xrge0infss ( 𝐶 ⊆ ( 0 [,] +∞ ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦𝐶 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧𝐶 𝑧 < 𝑦 ) ) )
14 12 13 syl ( 𝜑 → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦𝐶 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧𝐶 𝑧 < 𝑦 ) ) )
15 7 14 infcl ( 𝜑 → inf ( 𝐶 , ( 0 [,] +∞ ) , < ) ∈ ( 0 [,] +∞ ) )
16 3 15 sselid ( 𝜑 → inf ( 𝐶 , ( 0 [,] +∞ ) , < ) ∈ ℝ* )
17 7 1 14 9 infssd ( 𝜑 → ¬ inf ( 𝐶 , ( 0 [,] +∞ ) , < ) < inf ( 𝐵 , ( 0 [,] +∞ ) , < ) )
18 11 16 17 xrnltled ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ≤ inf ( 𝐶 , ( 0 [,] +∞ ) , < ) )