Metamath Proof Explorer


Theorem snunioo1

Description: The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion snunioo1 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 } ) = ( 𝐴 [,) 𝐵 ) )

Proof

Step Hyp Ref Expression
1 uncom ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐴 [,] 𝐴 ) ) = ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,) 𝐵 ) )
2 iccid ( 𝐴 ∈ ℝ* → ( 𝐴 [,] 𝐴 ) = { 𝐴 } )
3 2 3ad2ant1 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → ( 𝐴 [,] 𝐴 ) = { 𝐴 } )
4 3 uneq2d ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐴 [,] 𝐴 ) ) = ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 } ) )
5 simp1 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → 𝐴 ∈ ℝ* )
6 simp2 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → 𝐵 ∈ ℝ* )
7 xrleid ( 𝐴 ∈ ℝ*𝐴𝐴 )
8 7 3ad2ant1 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → 𝐴𝐴 )
9 simp3 ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → 𝐴 < 𝐵 )
10 df-icc [,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥𝑧𝑧𝑦 ) } )
11 df-ioo (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧𝑧 < 𝑦 ) } )
12 xrltnle ( ( 𝐴 ∈ ℝ*𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 ↔ ¬ 𝑤𝐴 ) )
13 df-ico [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥𝑧𝑧 < 𝑦 ) } )
14 xrlelttr ( ( 𝑤 ∈ ℝ*𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) → ( ( 𝑤𝐴𝐴 < 𝐵 ) → 𝑤 < 𝐵 ) )
15 simpl1 ( ( ( 𝐴 ∈ ℝ*𝐴 ∈ ℝ*𝑤 ∈ ℝ* ) ∧ ( 𝐴𝐴𝐴 < 𝑤 ) ) → 𝐴 ∈ ℝ* )
16 simpl3 ( ( ( 𝐴 ∈ ℝ*𝐴 ∈ ℝ*𝑤 ∈ ℝ* ) ∧ ( 𝐴𝐴𝐴 < 𝑤 ) ) → 𝑤 ∈ ℝ* )
17 simprr ( ( ( 𝐴 ∈ ℝ*𝐴 ∈ ℝ*𝑤 ∈ ℝ* ) ∧ ( 𝐴𝐴𝐴 < 𝑤 ) ) → 𝐴 < 𝑤 )
18 15 16 17 xrltled ( ( ( 𝐴 ∈ ℝ*𝐴 ∈ ℝ*𝑤 ∈ ℝ* ) ∧ ( 𝐴𝐴𝐴 < 𝑤 ) ) → 𝐴𝑤 )
19 18 ex ( ( 𝐴 ∈ ℝ*𝐴 ∈ ℝ*𝑤 ∈ ℝ* ) → ( ( 𝐴𝐴𝐴 < 𝑤 ) → 𝐴𝑤 ) )
20 10 11 12 13 14 19 ixxun ( ( ( 𝐴 ∈ ℝ*𝐴 ∈ ℝ*𝐵 ∈ ℝ* ) ∧ ( 𝐴𝐴𝐴 < 𝐵 ) ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 [,) 𝐵 ) )
21 5 5 6 8 9 20 syl32anc ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → ( ( 𝐴 [,] 𝐴 ) ∪ ( 𝐴 (,) 𝐵 ) ) = ( 𝐴 [,) 𝐵 ) )
22 1 4 21 3eqtr3a ( ( 𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 } ) = ( 𝐴 [,) 𝐵 ) )