Theorem snunioo

Description: The closure of one end of an open real interval. (Contributed by Paul Chapman, 15-Mar-2008) (Proof shortened by Mario Carneiro, 16-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ* )
2		iccid	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 [,] 𝐴 ) = { 𝐴 } )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( 𝐴 [,] 𝐴 ) = { 𝐴 } )
4	3	uneq1d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,) 𝐵 ) ) = ( { 𝐴 } ∪ ( 𝐴 (,) 𝐵 ) ) )
5		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* )
6	1	xrleidd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐴 )
7		simp3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 )
8		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
9		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
10		xrltnle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴 ) )
11		df-ico	⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } )
12		xrlelttr	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝑤 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) → 𝑤 < 𝐵 ) )
13		xrltle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 → 𝐴 ≤ 𝑤 ) )
14	13	3adant1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 → 𝐴 ≤ 𝑤 ) )
15	14	adantld	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) → 𝐴 ≤ 𝑤 ) )
16	8 9 10 11 12 15	ixxun	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 [,) 𝐵 ) )
17	1 1 5 6 7 16	syl32anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 [,) 𝐵 ) )
18	4 17	eqtr3d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → ( { 𝐴 } ∪ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 [,) 𝐵 ) )