Step | Hyp | Ref
| Expression |
1 | | simplr 768 |
. . . . 5
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → 𝑃 ≠ 1) |
2 | 1 | necomd 2996 |
. . . 4
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → 1 ≠ 𝑃) |
3 | | simpr 486 |
. . . . 5
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) |
4 | | nnz 12525 |
. . . . . . . 8
⊢ (𝑃 ∈ ℕ → 𝑃 ∈
ℤ) |
5 | | 1dvds 16158 |
. . . . . . . 8
⊢ (𝑃 ∈ ℤ → 1 ∥
𝑃) |
6 | 4, 5 | syl 17 |
. . . . . . 7
⊢ (𝑃 ∈ ℕ → 1 ∥
𝑃) |
7 | 6 | ad2antrr 725 |
. . . . . 6
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → 1 ∥
𝑃) |
8 | | 1nn 12169 |
. . . . . . 7
⊢ 1 ∈
ℕ |
9 | | breq1 5109 |
. . . . . . . 8
⊢ (𝑛 = 1 → (𝑛 ∥ 𝑃 ↔ 1 ∥ 𝑃)) |
10 | 9 | elrab3 3647 |
. . . . . . 7
⊢ (1 ∈
ℕ → (1 ∈ {𝑛
∈ ℕ ∣ 𝑛
∥ 𝑃} ↔ 1 ∥
𝑃)) |
11 | 8, 10 | ax-mp 5 |
. . . . . 6
⊢ (1 ∈
{𝑛 ∈ ℕ ∣
𝑛 ∥ 𝑃} ↔ 1 ∥ 𝑃) |
12 | 7, 11 | sylibr 233 |
. . . . 5
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → 1 ∈
{𝑛 ∈ ℕ ∣
𝑛 ∥ 𝑃}) |
13 | | iddvds 16157 |
. . . . . . . 8
⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 𝑃) |
14 | 4, 13 | syl 17 |
. . . . . . 7
⊢ (𝑃 ∈ ℕ → 𝑃 ∥ 𝑃) |
15 | 14 | ad2antrr 725 |
. . . . . 6
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → 𝑃 ∥ 𝑃) |
16 | | breq1 5109 |
. . . . . . . 8
⊢ (𝑛 = 𝑃 → (𝑛 ∥ 𝑃 ↔ 𝑃 ∥ 𝑃)) |
17 | 16 | elrab3 3647 |
. . . . . . 7
⊢ (𝑃 ∈ ℕ → (𝑃 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ↔ 𝑃 ∥ 𝑃)) |
18 | 17 | ad2antrr 725 |
. . . . . 6
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → (𝑃 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ↔ 𝑃 ∥ 𝑃)) |
19 | 15, 18 | mpbird 257 |
. . . . 5
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → 𝑃 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃}) |
20 | | en2eqpr 9948 |
. . . . 5
⊢ (({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o ∧ 1 ∈
{𝑛 ∈ ℕ ∣
𝑛 ∥ 𝑃} ∧ 𝑃 ∈ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃}) → (1 ≠ 𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})) |
21 | 3, 12, 19, 20 | syl3anc 1372 |
. . . 4
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → (1 ≠
𝑃 → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})) |
22 | 2, 21 | mpd 15 |
. . 3
⊢ (((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) ∧ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o) → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃}) |
23 | 22 | ex 414 |
. 2
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})) |
24 | | necom 2994 |
. . . 4
⊢ (1 ≠
𝑃 ↔ 𝑃 ≠ 1) |
25 | | pr2ne 9945 |
. . . . . 6
⊢ ((1
∈ ℕ ∧ 𝑃
∈ ℕ) → ({1, 𝑃} ≈ 2o ↔ 1 ≠ 𝑃)) |
26 | 8, 25 | mpan 689 |
. . . . 5
⊢ (𝑃 ∈ ℕ → ({1,
𝑃} ≈ 2o
↔ 1 ≠ 𝑃)) |
27 | 26 | biimpar 479 |
. . . 4
⊢ ((𝑃 ∈ ℕ ∧ 1 ≠
𝑃) → {1, 𝑃} ≈
2o) |
28 | 24, 27 | sylan2br 596 |
. . 3
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → {1, 𝑃} ≈
2o) |
29 | | breq1 5109 |
. . 3
⊢ ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃} → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o ↔ {1, 𝑃} ≈
2o)) |
30 | 28, 29 | syl5ibrcom 247 |
. 2
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃} → {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o)) |
31 | 23, 30 | impbid 211 |
1
⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ≠ 1) → ({𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} ≈ 2o ↔ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑃} = {1, 𝑃})) |