| Step |
Hyp |
Ref |
Expression |
| 1 |
|
3wlkd.p |
⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ |
| 2 |
|
3wlkd.f |
⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩ |
| 3 |
|
3wlkd.s |
⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) ) |
| 4 |
|
3wlkd.n |
⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) |
| 5 |
1 2 3
|
3wlkdlem3 |
⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ) |
| 6 |
|
simpr1l |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → 𝐴 ≠ 𝐵 ) |
| 7 |
|
simpl |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 0 ) = 𝐴 ) |
| 8 |
7
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 0 ) = 𝐴 ) |
| 9 |
|
simpr |
⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( 𝑃 ‘ 1 ) = 𝐵 ) |
| 10 |
9
|
adantr |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 1 ) = 𝐵 ) |
| 11 |
8 10
|
neeq12d |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ↔ 𝐴 ≠ 𝐵 ) ) |
| 12 |
11
|
adantr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ↔ 𝐴 ≠ 𝐵 ) ) |
| 13 |
6 12
|
mpbird |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) |
| 14 |
13
|
a1d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 15 |
|
simpr1r |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → 𝐴 ≠ 𝐶 ) |
| 16 |
|
simpl |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( 𝑃 ‘ 2 ) = 𝐶 ) |
| 17 |
16
|
adantl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 2 ) = 𝐶 ) |
| 18 |
8 17
|
neeq12d |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐴 ≠ 𝐶 ) ) |
| 19 |
18
|
adantr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐴 ≠ 𝐶 ) ) |
| 20 |
15 19
|
mpbird |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) |
| 21 |
20
|
a1d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 22 |
14 21
|
jca |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 23 |
|
eqid |
⊢ 1 = 1 |
| 24 |
23
|
2a1i |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) = ( 𝑃 ‘ 1 ) → 1 = 1 ) ) |
| 25 |
24
|
necon3d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 26 |
|
simpr2l |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → 𝐵 ≠ 𝐶 ) |
| 27 |
10 17
|
neeq12d |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐵 ≠ 𝐶 ) ) |
| 28 |
27
|
adantr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐵 ≠ 𝐶 ) ) |
| 29 |
26 28
|
mpbird |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) |
| 30 |
29
|
a1d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 31 |
25 30
|
jca |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 32 |
29
|
necomd |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) |
| 33 |
32
|
a1d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 34 |
|
eqid |
⊢ 2 = 2 |
| 35 |
34
|
2a1i |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 2 ) = ( 𝑃 ‘ 2 ) → 2 = 2 ) ) |
| 36 |
35
|
necon3d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 37 |
|
simpr2r |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → 𝐵 ≠ 𝐷 ) |
| 38 |
|
simpr |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( 𝑃 ‘ 3 ) = 𝐷 ) |
| 39 |
38
|
adantl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( 𝑃 ‘ 3 ) = 𝐷 ) |
| 40 |
10 39
|
neeq12d |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 3 ) ↔ 𝐵 ≠ 𝐷 ) ) |
| 41 |
40
|
adantr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 3 ) ↔ 𝐵 ≠ 𝐷 ) ) |
| 42 |
37 41
|
mpbird |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 3 ) ) |
| 43 |
42
|
necomd |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) |
| 44 |
43
|
a1d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 45 |
|
simp3 |
⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) → 𝐶 ≠ 𝐷 ) |
| 46 |
45
|
necomd |
⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) → 𝐷 ≠ 𝐶 ) |
| 47 |
46
|
adantl |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → 𝐷 ≠ 𝐶 ) |
| 48 |
|
simpl |
⊢ ( ( ( 𝑃 ‘ 3 ) = 𝐷 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 3 ) = 𝐷 ) |
| 49 |
|
simpr |
⊢ ( ( ( 𝑃 ‘ 3 ) = 𝐷 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( 𝑃 ‘ 2 ) = 𝐶 ) |
| 50 |
48 49
|
neeq12d |
⊢ ( ( ( 𝑃 ‘ 3 ) = 𝐷 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → ( ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐷 ≠ 𝐶 ) ) |
| 51 |
50
|
ancoms |
⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐷 ≠ 𝐶 ) ) |
| 52 |
51
|
adantl |
⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐷 ≠ 𝐶 ) ) |
| 53 |
52
|
adantr |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ↔ 𝐷 ≠ 𝐶 ) ) |
| 54 |
47 53
|
mpbird |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) |
| 55 |
54
|
a1d |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 56 |
44 55
|
jca |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 57 |
33 36 56
|
jca31 |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 58 |
22 31 57
|
jca31 |
⊢ ( ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) ∧ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) ) → ( ( ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ∧ ( ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) ) |
| 59 |
5 4 58
|
syl2anc |
⊢ ( 𝜑 → ( ( ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ∧ ( ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) ) |
| 60 |
1
|
fveq2i |
⊢ ( ♯ ‘ 𝑃 ) = ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ) |
| 61 |
|
s4len |
⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ) = 4 |
| 62 |
60 61
|
eqtri |
⊢ ( ♯ ‘ 𝑃 ) = 4 |
| 63 |
62
|
oveq2i |
⊢ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) = ( 0 ..^ 4 ) |
| 64 |
|
fzo0to42pr |
⊢ ( 0 ..^ 4 ) = ( { 0 , 1 } ∪ { 2 , 3 } ) |
| 65 |
63 64
|
eqtri |
⊢ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) = ( { 0 , 1 } ∪ { 2 , 3 } ) |
| 66 |
65
|
raleqi |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 67 |
|
ralunb |
⊢ ( ∀ 𝑘 ∈ ( { 0 , 1 } ∪ { 2 , 3 } ) ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ∀ 𝑘 ∈ { 0 , 1 } ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ∀ 𝑘 ∈ { 2 , 3 } ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 68 |
|
c0ex |
⊢ 0 ∈ V |
| 69 |
|
1ex |
⊢ 1 ∈ V |
| 70 |
|
neeq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 ≠ 1 ↔ 0 ≠ 1 ) ) |
| 71 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 0 ) ) |
| 72 |
71
|
neeq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 73 |
70 72
|
imbi12d |
⊢ ( 𝑘 = 0 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ) ) |
| 74 |
|
neeq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 ≠ 2 ↔ 0 ≠ 2 ) ) |
| 75 |
71
|
neeq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 76 |
74 75
|
imbi12d |
⊢ ( 𝑘 = 0 → ( ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ↔ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 77 |
73 76
|
anbi12d |
⊢ ( 𝑘 = 0 → ( ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 78 |
|
neeq1 |
⊢ ( 𝑘 = 1 → ( 𝑘 ≠ 1 ↔ 1 ≠ 1 ) ) |
| 79 |
|
fveq2 |
⊢ ( 𝑘 = 1 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 1 ) ) |
| 80 |
79
|
neeq1d |
⊢ ( 𝑘 = 1 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 81 |
78 80
|
imbi12d |
⊢ ( 𝑘 = 1 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ) ) |
| 82 |
|
neeq1 |
⊢ ( 𝑘 = 1 → ( 𝑘 ≠ 2 ↔ 1 ≠ 2 ) ) |
| 83 |
79
|
neeq1d |
⊢ ( 𝑘 = 1 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ↔ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 84 |
82 83
|
imbi12d |
⊢ ( 𝑘 = 1 → ( ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ↔ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 85 |
81 84
|
anbi12d |
⊢ ( 𝑘 = 1 → ( ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 86 |
68 69 77 85
|
ralpr |
⊢ ( ∀ 𝑘 ∈ { 0 , 1 } ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 87 |
|
2ex |
⊢ 2 ∈ V |
| 88 |
|
3ex |
⊢ 3 ∈ V |
| 89 |
|
neeq1 |
⊢ ( 𝑘 = 2 → ( 𝑘 ≠ 1 ↔ 2 ≠ 1 ) ) |
| 90 |
|
fveq2 |
⊢ ( 𝑘 = 2 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 2 ) ) |
| 91 |
90
|
neeq1d |
⊢ ( 𝑘 = 2 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 92 |
89 91
|
imbi12d |
⊢ ( 𝑘 = 2 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ) ) |
| 93 |
|
neeq1 |
⊢ ( 𝑘 = 2 → ( 𝑘 ≠ 2 ↔ 2 ≠ 2 ) ) |
| 94 |
90
|
neeq1d |
⊢ ( 𝑘 = 2 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ↔ ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 95 |
93 94
|
imbi12d |
⊢ ( 𝑘 = 2 → ( ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ↔ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 96 |
92 95
|
anbi12d |
⊢ ( 𝑘 = 2 → ( ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 97 |
|
neeq1 |
⊢ ( 𝑘 = 3 → ( 𝑘 ≠ 1 ↔ 3 ≠ 1 ) ) |
| 98 |
|
fveq2 |
⊢ ( 𝑘 = 3 → ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ 3 ) ) |
| 99 |
98
|
neeq1d |
⊢ ( 𝑘 = 3 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ↔ ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 100 |
97 99
|
imbi12d |
⊢ ( 𝑘 = 3 → ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ↔ ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ) ) |
| 101 |
|
neeq1 |
⊢ ( 𝑘 = 3 → ( 𝑘 ≠ 2 ↔ 3 ≠ 2 ) ) |
| 102 |
98
|
neeq1d |
⊢ ( 𝑘 = 3 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ↔ ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 103 |
101 102
|
imbi12d |
⊢ ( 𝑘 = 3 → ( ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ↔ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 104 |
100 103
|
anbi12d |
⊢ ( 𝑘 = 3 → ( ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 105 |
87 88 96 104
|
ralpr |
⊢ ( ∀ 𝑘 ∈ { 2 , 3 } ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) |
| 106 |
86 105
|
anbi12i |
⊢ ( ( ∀ 𝑘 ∈ { 0 , 1 } ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ∀ 𝑘 ∈ { 2 , 3 } ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ↔ ( ( ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ∧ ( ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) ) |
| 107 |
66 67 106
|
3bitri |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ↔ ( ( ( ( 0 ≠ 1 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 0 ≠ 2 → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 1 ≠ 1 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 1 ≠ 2 → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ∧ ( ( ( 2 ≠ 1 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 2 ≠ 2 → ( 𝑃 ‘ 2 ) ≠ ( 𝑃 ‘ 2 ) ) ) ∧ ( ( 3 ≠ 1 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 3 ≠ 2 → ( 𝑃 ‘ 3 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) ) ) |
| 108 |
59 107
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 109 |
2
|
fveq2i |
⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ) |
| 110 |
|
s3len |
⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 𝐿 ”⟩ ) = 3 |
| 111 |
109 110
|
eqtri |
⊢ ( ♯ ‘ 𝐹 ) = 3 |
| 112 |
111
|
oveq2i |
⊢ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) = ( 1 ..^ 3 ) |
| 113 |
|
fzo13pr |
⊢ ( 1 ..^ 3 ) = { 1 , 2 } |
| 114 |
112 113
|
eqtri |
⊢ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) = { 1 , 2 } |
| 115 |
114
|
raleqi |
⊢ ( ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ∀ 𝑗 ∈ { 1 , 2 } ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ) |
| 116 |
|
neeq2 |
⊢ ( 𝑗 = 1 → ( 𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 1 ) ) |
| 117 |
|
fveq2 |
⊢ ( 𝑗 = 1 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 1 ) ) |
| 118 |
117
|
neeq2d |
⊢ ( 𝑗 = 1 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ↔ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ) |
| 119 |
116 118
|
imbi12d |
⊢ ( 𝑗 = 1 → ( ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ) ) |
| 120 |
|
neeq2 |
⊢ ( 𝑗 = 2 → ( 𝑘 ≠ 𝑗 ↔ 𝑘 ≠ 2 ) ) |
| 121 |
|
fveq2 |
⊢ ( 𝑗 = 2 → ( 𝑃 ‘ 𝑗 ) = ( 𝑃 ‘ 2 ) ) |
| 122 |
121
|
neeq2d |
⊢ ( 𝑗 = 2 → ( ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ↔ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) |
| 123 |
120 122
|
imbi12d |
⊢ ( 𝑗 = 2 → ( ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 124 |
69 87 119 123
|
ralpr |
⊢ ( ∀ 𝑗 ∈ { 1 , 2 } ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 125 |
115 124
|
bitri |
⊢ ( ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 126 |
125
|
ralbii |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ↔ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ( ( 𝑘 ≠ 1 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 1 ) ) ∧ ( 𝑘 ≠ 2 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 2 ) ) ) ) |
| 127 |
108 126
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑃 ) ) ∀ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝑘 ≠ 𝑗 → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ 𝑗 ) ) ) |