Step |
Hyp |
Ref |
Expression |
1 |
|
uzfbas.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
1
|
uzrest |
⊢ ( 𝑀 ∈ ℤ → ( ran ℤ≥ ↾t 𝑍 ) = ( ℤ≥ “ 𝑍 ) ) |
3 |
|
zfbas |
⊢ ran ℤ≥ ∈ ( fBas ‘ ℤ ) |
4 |
|
0nelfb |
⊢ ( ran ℤ≥ ∈ ( fBas ‘ ℤ ) → ¬ ∅ ∈ ran ℤ≥ ) |
5 |
3 4
|
ax-mp |
⊢ ¬ ∅ ∈ ran ℤ≥ |
6 |
|
imassrn |
⊢ ( ℤ≥ “ 𝑍 ) ⊆ ran ℤ≥ |
7 |
2 6
|
eqsstrdi |
⊢ ( 𝑀 ∈ ℤ → ( ran ℤ≥ ↾t 𝑍 ) ⊆ ran ℤ≥ ) |
8 |
7
|
sseld |
⊢ ( 𝑀 ∈ ℤ → ( ∅ ∈ ( ran ℤ≥ ↾t 𝑍 ) → ∅ ∈ ran ℤ≥ ) ) |
9 |
5 8
|
mtoi |
⊢ ( 𝑀 ∈ ℤ → ¬ ∅ ∈ ( ran ℤ≥ ↾t 𝑍 ) ) |
10 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
11 |
1 10
|
eqsstri |
⊢ 𝑍 ⊆ ℤ |
12 |
|
trfbas2 |
⊢ ( ( ran ℤ≥ ∈ ( fBas ‘ ℤ ) ∧ 𝑍 ⊆ ℤ ) → ( ( ran ℤ≥ ↾t 𝑍 ) ∈ ( fBas ‘ 𝑍 ) ↔ ¬ ∅ ∈ ( ran ℤ≥ ↾t 𝑍 ) ) ) |
13 |
3 11 12
|
mp2an |
⊢ ( ( ran ℤ≥ ↾t 𝑍 ) ∈ ( fBas ‘ 𝑍 ) ↔ ¬ ∅ ∈ ( ran ℤ≥ ↾t 𝑍 ) ) |
14 |
9 13
|
sylibr |
⊢ ( 𝑀 ∈ ℤ → ( ran ℤ≥ ↾t 𝑍 ) ∈ ( fBas ‘ 𝑍 ) ) |
15 |
2 14
|
eqeltrrd |
⊢ ( 𝑀 ∈ ℤ → ( ℤ≥ “ 𝑍 ) ∈ ( fBas ‘ 𝑍 ) ) |