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 ) ) |