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Metamath Proof Explorer

Theorem icopnfsup

Description: An upper set of reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016)

Ref Expression
Assertion icopnfsup ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → sup ( ( 𝐴 [,) +∞ ) , ℝ* , < ) = +∞ )

Proof

Step Hyp Ref Expression
1 simpl ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → 𝐴 ∈ ℝ* )
2 pnfxr ⊢ +∞ ∈ ℝ*
3 2 a1i ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → +∞ ∈ ℝ* )
4 nltpnft ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 = +∞ ↔ ¬ 𝐴 < +∞ ) )
5 4 necon2abid ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 < +∞ ↔ 𝐴 ≠ +∞ ) )
6 5 biimpar ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → 𝐴 < +∞ )
7 lbico1 ⊢ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 < +∞ ) → 𝐴 ∈ ( 𝐴 [,) +∞ ) )
8 1 3 6 7 syl3anc ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → 𝐴 ∈ ( 𝐴 [,) +∞ ) )
9 8 ne0d ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → ( 𝐴 [,) +∞ ) ≠ ∅ )
10 df-ico ⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } )
11 idd ⊢ ( ( 𝑤 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑤 < +∞ → 𝑤 < +∞ ) )
12 xrltle ⊢ ( ( 𝑤 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑤 < +∞ → 𝑤 ≤ +∞ ) )
13 xrltle ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 → 𝐴 ≤ 𝑤 ) )
14 idd ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 ≤ 𝑤 → 𝐴 ≤ 𝑤 ) )
15 10 11 12 13 14 ixxub ⊢ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ( 𝐴 [,) +∞ ) ≠ ∅ ) → sup ( ( 𝐴 [,) +∞ ) , ℝ* , < ) = +∞ )
16 1 3 9 15 syl3anc ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞ ) → sup ( ( 𝐴 [,) +∞ ) , ℝ* , < ) = +∞ )