Step |
Hyp |
Ref |
Expression |
1 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝐾 ... 𝑁 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝐾 ) ) |
2 |
|
id |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
3 |
|
uztrn |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝐾 ) ∧ 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
4 |
1 2 3
|
syl2anr |
⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑘 ∈ ( 𝐾 ... 𝑁 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
5 |
|
elfzuz3 |
⊢ ( 𝑘 ∈ ( 𝐾 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑘 ∈ ( 𝐾 ... 𝑁 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) ) |
7 |
|
elfzuzb |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) ) ) |
8 |
4 6 7
|
sylanbrc |
⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑘 ∈ ( 𝐾 ... 𝑁 ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
9 |
8
|
ex |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑘 ∈ ( 𝐾 ... 𝑁 ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ) |
10 |
9
|
ssrdv |
⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) ) |