Step |
Hyp |
Ref |
Expression |
1 |
|
leordtval.1 |
⊢ 𝐴 = ran ( 𝑥 ∈ ℝ* ↦ ( 𝑥 (,] +∞ ) ) |
2 |
|
iocssxr |
⊢ ( 𝑥 (,] +∞ ) ⊆ ℝ* |
3 |
|
sseqin2 |
⊢ ( ( 𝑥 (,] +∞ ) ⊆ ℝ* ↔ ( ℝ* ∩ ( 𝑥 (,] +∞ ) ) = ( 𝑥 (,] +∞ ) ) |
4 |
2 3
|
mpbi |
⊢ ( ℝ* ∩ ( 𝑥 (,] +∞ ) ) = ( 𝑥 (,] +∞ ) |
5 |
|
simpl |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → 𝑥 ∈ ℝ* ) |
6 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
7 |
|
elioc1 |
⊢ ( ( 𝑥 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑦 ∈ ( 𝑥 (,] +∞ ) ↔ ( 𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞ ) ) ) |
8 |
5 6 7
|
sylancl |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑦 ∈ ( 𝑥 (,] +∞ ) ↔ ( 𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞ ) ) ) |
9 |
|
simpr |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → 𝑦 ∈ ℝ* ) |
10 |
|
pnfge |
⊢ ( 𝑦 ∈ ℝ* → 𝑦 ≤ +∞ ) |
11 |
9 10
|
jccir |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞ ) ) |
12 |
11
|
biantrurd |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 < 𝑦 ↔ ( ( 𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞ ) ∧ 𝑥 < 𝑦 ) ) ) |
13 |
|
3anan32 |
⊢ ( ( 𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞ ) ↔ ( ( 𝑦 ∈ ℝ* ∧ 𝑦 ≤ +∞ ) ∧ 𝑥 < 𝑦 ) ) |
14 |
12 13
|
bitr4di |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 < 𝑦 ↔ ( 𝑦 ∈ ℝ* ∧ 𝑥 < 𝑦 ∧ 𝑦 ≤ +∞ ) ) ) |
15 |
|
xrltnle |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 < 𝑦 ↔ ¬ 𝑦 ≤ 𝑥 ) ) |
16 |
8 14 15
|
3bitr2d |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑦 ∈ ( 𝑥 (,] +∞ ) ↔ ¬ 𝑦 ≤ 𝑥 ) ) |
17 |
16
|
rabbi2dva |
⊢ ( 𝑥 ∈ ℝ* → ( ℝ* ∩ ( 𝑥 (,] +∞ ) ) = { 𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥 } ) |
18 |
4 17
|
syl5eqr |
⊢ ( 𝑥 ∈ ℝ* → ( 𝑥 (,] +∞ ) = { 𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥 } ) |
19 |
18
|
mpteq2ia |
⊢ ( 𝑥 ∈ ℝ* ↦ ( 𝑥 (,] +∞ ) ) = ( 𝑥 ∈ ℝ* ↦ { 𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥 } ) |
20 |
19
|
rneqi |
⊢ ran ( 𝑥 ∈ ℝ* ↦ ( 𝑥 (,] +∞ ) ) = ran ( 𝑥 ∈ ℝ* ↦ { 𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥 } ) |
21 |
1 20
|
eqtri |
⊢ 𝐴 = ran ( 𝑥 ∈ ℝ* ↦ { 𝑦 ∈ ℝ* ∣ ¬ 𝑦 ≤ 𝑥 } ) |