Step |
Hyp |
Ref |
Expression |
1 |
|
0ne1 |
⊢ 0 ≠ 1 |
2 |
|
0re |
⊢ 0 ∈ ℝ |
3 |
|
1re |
⊢ 1 ∈ ℝ |
4 |
2 3
|
lttri2i |
⊢ ( 0 ≠ 1 ↔ ( 0 < 1 ∨ 1 < 0 ) ) |
5 |
1 4
|
mpbi |
⊢ ( 0 < 1 ∨ 1 < 0 ) |
6 |
|
breq2 |
⊢ ( 𝑥 = 1 → ( 0 < 𝑥 ↔ 0 < 1 ) ) |
7 |
|
breq2 |
⊢ ( 𝑥 = 𝑦 → ( 0 < 𝑥 ↔ 0 < 𝑦 ) ) |
8 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 0 < 𝑥 ↔ 0 < ( 𝑦 + 1 ) ) ) |
9 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( 0 < 𝑥 ↔ 0 < 𝐴 ) ) |
10 |
|
id |
⊢ ( 0 < 1 → 0 < 1 ) |
11 |
|
nnre |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℝ ) |
12 |
11
|
ad2antlr |
⊢ ( ( ( 0 < 1 ∧ 𝑦 ∈ ℕ ) ∧ 0 < 𝑦 ) → 𝑦 ∈ ℝ ) |
13 |
|
1red |
⊢ ( ( ( 0 < 1 ∧ 𝑦 ∈ ℕ ) ∧ 0 < 𝑦 ) → 1 ∈ ℝ ) |
14 |
|
simpr |
⊢ ( ( ( 0 < 1 ∧ 𝑦 ∈ ℕ ) ∧ 0 < 𝑦 ) → 0 < 𝑦 ) |
15 |
|
simpll |
⊢ ( ( ( 0 < 1 ∧ 𝑦 ∈ ℕ ) ∧ 0 < 𝑦 ) → 0 < 1 ) |
16 |
12 13 14 15
|
sn-addgt0d |
⊢ ( ( ( 0 < 1 ∧ 𝑦 ∈ ℕ ) ∧ 0 < 𝑦 ) → 0 < ( 𝑦 + 1 ) ) |
17 |
6 7 8 9 10 16
|
nnindd |
⊢ ( ( 0 < 1 ∧ 𝐴 ∈ ℕ ) → 0 < 𝐴 ) |
18 |
17
|
gt0ne0d |
⊢ ( ( 0 < 1 ∧ 𝐴 ∈ ℕ ) → 𝐴 ≠ 0 ) |
19 |
18
|
ancoms |
⊢ ( ( 𝐴 ∈ ℕ ∧ 0 < 1 ) → 𝐴 ≠ 0 ) |
20 |
|
breq1 |
⊢ ( 𝑥 = 1 → ( 𝑥 < 0 ↔ 1 < 0 ) ) |
21 |
|
breq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 < 0 ↔ 𝑦 < 0 ) ) |
22 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 < 0 ↔ ( 𝑦 + 1 ) < 0 ) ) |
23 |
|
breq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 < 0 ↔ 𝐴 < 0 ) ) |
24 |
|
id |
⊢ ( 1 < 0 → 1 < 0 ) |
25 |
11
|
ad2antlr |
⊢ ( ( ( 1 < 0 ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 0 ) → 𝑦 ∈ ℝ ) |
26 |
|
1red |
⊢ ( ( ( 1 < 0 ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 0 ) → 1 ∈ ℝ ) |
27 |
|
simpr |
⊢ ( ( ( 1 < 0 ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 0 ) → 𝑦 < 0 ) |
28 |
|
simpll |
⊢ ( ( ( 1 < 0 ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 0 ) → 1 < 0 ) |
29 |
25 26 27 28
|
sn-addlt0d |
⊢ ( ( ( 1 < 0 ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 0 ) → ( 𝑦 + 1 ) < 0 ) |
30 |
20 21 22 23 24 29
|
nnindd |
⊢ ( ( 1 < 0 ∧ 𝐴 ∈ ℕ ) → 𝐴 < 0 ) |
31 |
30
|
lt0ne0d |
⊢ ( ( 1 < 0 ∧ 𝐴 ∈ ℕ ) → 𝐴 ≠ 0 ) |
32 |
31
|
ancoms |
⊢ ( ( 𝐴 ∈ ℕ ∧ 1 < 0 ) → 𝐴 ≠ 0 ) |
33 |
19 32
|
jaodan |
⊢ ( ( 𝐴 ∈ ℕ ∧ ( 0 < 1 ∨ 1 < 0 ) ) → 𝐴 ≠ 0 ) |
34 |
5 33
|
mpan2 |
⊢ ( 𝐴 ∈ ℕ → 𝐴 ≠ 0 ) |