Step |
Hyp |
Ref |
Expression |
1 |
|
incom |
⊢ ( { 𝑀 } ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∩ { 𝑀 } ) |
2 |
|
0lt1 |
⊢ 0 < 1 |
3 |
|
0re |
⊢ 0 ∈ ℝ |
4 |
|
1re |
⊢ 1 ∈ ℝ |
5 |
3 4
|
ltnlei |
⊢ ( 0 < 1 ↔ ¬ 1 ≤ 0 ) |
6 |
2 5
|
mpbi |
⊢ ¬ 1 ≤ 0 |
7 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
8 |
7
|
zred |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℝ ) |
9 |
|
leaddle0 |
⊢ ( ( 𝑀 ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 𝑀 + 1 ) ≤ 𝑀 ↔ 1 ≤ 0 ) ) |
10 |
8 4 9
|
sylancl |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 + 1 ) ≤ 𝑀 ↔ 1 ≤ 0 ) ) |
11 |
6 10
|
mtbiri |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ ( 𝑀 + 1 ) ≤ 𝑀 ) |
12 |
11
|
intnanrd |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ ( ( 𝑀 + 1 ) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁 ) ) |
13 |
12
|
intnand |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ ( ( ( 𝑀 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( ( 𝑀 + 1 ) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁 ) ) ) |
14 |
|
elfz2 |
⊢ ( 𝑀 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ↔ ( ( ( 𝑀 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( ( 𝑀 + 1 ) ≤ 𝑀 ∧ 𝑀 ≤ 𝑁 ) ) ) |
15 |
13 14
|
sylnibr |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ 𝑀 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
16 |
|
disjsn |
⊢ ( ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∩ { 𝑀 } ) = ∅ ↔ ¬ 𝑀 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) ) |
17 |
15 16
|
sylibr |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( ( 𝑀 + 1 ) ... 𝑁 ) ∩ { 𝑀 } ) = ∅ ) |
18 |
1 17
|
eqtrid |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( { 𝑀 } ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ ) |