Theorem ioounsn

Description: The union of an open interval with its upper endpoint is a left-open right-closed interval. (Contributed by Jon Pennant, 8-Jun-2019)

	Ref	Expression
Assertion	ioounsn	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) = ( 𝐴 (,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* )
2		iccid	⊢ ( 𝐵 ∈ ℝ* → ( 𝐵 [,] 𝐵 ) = { 𝐵 } )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( 𝐵 [,] 𝐵 ) = { 𝐵 } )
4	3	uneq2d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) )
5		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ* )
6		simp3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 )
7	1	xrleidd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐵 )
8		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
9		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
10		xrlenlt	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵 ) )
11		df-ioc	⊢ (,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
12		simpl1	⊢ ( ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝑤 ∈ ℝ* )
13		simpl2	⊢ ( ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝐵 ∈ ℝ* )
14		simprl	⊢ ( ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝑤 < 𝐵 )
15	12 13 14	xrltled	⊢ ( ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝑤 ≤ 𝐵 )
16	15	ex	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) → 𝑤 ≤ 𝐵 ) )
17		xrltletr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤 ) → 𝐴 < 𝑤 ) )
18	8 9 10 11 16 17	ixxun	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( 𝐴 (,] 𝐵 ) )
19	5 1 1 6 7 18	syl32anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( 𝐴 (,] 𝐵 ) )
20	4 19	eqtr3d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) = ( 𝐴 (,] 𝐵 ) )