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* , < ) = +∞ )