Theorem ioojoin

Description: Join two open intervals to create a third. (Contributed by NM, 11-Aug-2008) (Proof shortened by Mario Carneiro, 16-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		unass	⊢ ( ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) ∪ ( 𝐵 (,) 𝐶 ) ) = ( ( 𝐴 (,) 𝐵 ) ∪ ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) )
2		snunioo	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 < 𝐶 ) → ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) = ( 𝐵 [,) 𝐶 ) )
3	2	3expa	⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐵 < 𝐶 ) → ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) = ( 𝐵 [,) 𝐶 ) )
4	3	3adantl1	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐵 < 𝐶 ) → ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) = ( 𝐵 [,) 𝐶 ) )
5	4	adantrl	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ) → ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) = ( 𝐵 [,) 𝐶 ) )
6	5	uneq2d	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) ) = ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,) 𝐶 ) ) )
7		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
8		df-ico	⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } )
9		xrlenlt	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵 ) )
10		xrlttr	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝑤 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝑤 < 𝐶 ) )
11		xrltletr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤 ) → 𝐴 < 𝑤 ) )
12	7 8 9 7 10 11	ixxun	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,) 𝐶 ) ) = ( 𝐴 (,) 𝐶 ) )
13	6 12	eqtrd	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( { 𝐵 } ∪ ( 𝐵 (,) 𝐶 ) ) ) = ( 𝐴 (,) 𝐶 ) )
14	1 13	eqtrid	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) ) → ( ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) ∪ ( 𝐵 (,) 𝐶 ) ) = ( 𝐴 (,) 𝐶 ) )