Theorem snunioc

Description: The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017)

	Ref	Expression
Assertion	snunioc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( { 𝐴 } ∪ ( 𝐴 (,] 𝐵 ) ) = ( 𝐴 [,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		iccid	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 [,] 𝐴 ) = { 𝐴 } )
2	1	3ad2ant1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 [,] 𝐴 ) = { 𝐴 } )
3	2	uneq1d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,] 𝐵 ) ) = ( { 𝐴 } ∪ ( 𝐴 (,] 𝐵 ) ) )
4		simp1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ℝ* )
5		simp2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ℝ* )
6		xrleid	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )
7	6	3ad2ant1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐴 )
8		simp3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 )
9		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
10		df-ioc	⊢ (,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
11		xrltnle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴 ) )
12		xrletr	⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝑤 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → 𝑤 ≤ 𝐵 ) )
13		simpl1	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝐴 ∈ ℝ* )
14		simpl3	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝑤 ∈ ℝ* )
15		simprr	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝐴 < 𝑤 )
16	13 14 15	xrltled	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝐴 ≤ 𝑤 )
17	16	ex	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) → 𝐴 ≤ 𝑤 ) )
18	9 10 11 9 12 17	ixxun	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,] 𝐵 ) ) = ( 𝐴 [,] 𝐵 ) )
19	4 4 5 7 8 18	syl32anc	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,] 𝐵 ) ) = ( 𝐴 [,] 𝐵 ) )
20	3 19	eqtr3d	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( { 𝐴 } ∪ ( 𝐴 (,] 𝐵 ) ) = ( 𝐴 [,] 𝐵 ) )