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* → 𝐴 ≤ 𝐴 ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐴 ) |
9 |
|
simp3 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 ) |
10 |
|
df-icc |
⊢ [,] = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } ) |
11 |
|
df-ioo |
⊢ (,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } ) |
12 |
|
xrltnle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( 𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴 ) ) |
13 |
|
df-ico |
⊢ [,) = ( 𝑥 ∈ ℝ* , 𝑦 ∈ ℝ* ↦ { 𝑧 ∈ ℝ* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦 ) } ) |
14 |
|
xrlelttr |
⊢ ( ( 𝑤 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝑤 ≤ 𝐴 ∧ 𝐴 < 𝐵 ) → 𝑤 < 𝐵 ) ) |
15 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝐴 ∈ ℝ* ) |
16 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝑤 ∈ ℝ* ) |
17 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝐴 < 𝑤 ) |
18 |
15 16 17
|
xrltled |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) ) → 𝐴 ≤ 𝑤 ) |
19 |
18
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑤 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝑤 ) → 𝐴 ≤ 𝑤 ) ) |
20 |
10 11 12 13 14 19
|
ixxun |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐴 ∧ 𝐴 < 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 ) ) |