Step |
Hyp |
Ref |
Expression |
1 |
|
konigsberg.v |
⊢ 𝑉 = ( 0 ... 3 ) |
2 |
|
konigsberg.e |
⊢ 𝐸 = 〈“ { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ”〉 |
3 |
|
konigsberg.g |
⊢ 𝐺 = 〈 𝑉 , 𝐸 〉 |
4 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
5 |
|
0elfz |
⊢ ( 3 ∈ ℕ0 → 0 ∈ ( 0 ... 3 ) ) |
6 |
4 5
|
ax-mp |
⊢ 0 ∈ ( 0 ... 3 ) |
7 |
6 1
|
eleqtrri |
⊢ 0 ∈ 𝑉 |
8 |
|
n2dvds3 |
⊢ ¬ 2 ∥ 3 |
9 |
1 2 3
|
konigsberglem1 |
⊢ ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) = 3 |
10 |
9
|
breq2i |
⊢ ( 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) ↔ 2 ∥ 3 ) |
11 |
8 10
|
mtbir |
⊢ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) |
12 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) = ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) ) |
13 |
12
|
breq2d |
⊢ ( 𝑥 = 0 → ( 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) ↔ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) ) ) |
14 |
13
|
notbid |
⊢ ( 𝑥 = 0 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) ↔ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) ) ) |
15 |
14
|
elrab |
⊢ ( 0 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ↔ ( 0 ∈ 𝑉 ∧ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 0 ) ) ) |
16 |
7 11 15
|
mpbir2an |
⊢ 0 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } |
17 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
18 |
|
1le3 |
⊢ 1 ≤ 3 |
19 |
|
elfz2nn0 |
⊢ ( 1 ∈ ( 0 ... 3 ) ↔ ( 1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3 ) ) |
20 |
17 4 18 19
|
mpbir3an |
⊢ 1 ∈ ( 0 ... 3 ) |
21 |
20 1
|
eleqtrri |
⊢ 1 ∈ 𝑉 |
22 |
1 2 3
|
konigsberglem2 |
⊢ ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) = 3 |
23 |
22
|
breq2i |
⊢ ( 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) ↔ 2 ∥ 3 ) |
24 |
8 23
|
mtbir |
⊢ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) |
25 |
|
fveq2 |
⊢ ( 𝑥 = 1 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) = ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) ) |
26 |
25
|
breq2d |
⊢ ( 𝑥 = 1 → ( 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) ↔ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) ) ) |
27 |
26
|
notbid |
⊢ ( 𝑥 = 1 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) ↔ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) ) ) |
28 |
27
|
elrab |
⊢ ( 1 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ↔ ( 1 ∈ 𝑉 ∧ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 1 ) ) ) |
29 |
21 24 28
|
mpbir2an |
⊢ 1 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } |
30 |
|
3re |
⊢ 3 ∈ ℝ |
31 |
30
|
leidi |
⊢ 3 ≤ 3 |
32 |
|
elfz2nn0 |
⊢ ( 3 ∈ ( 0 ... 3 ) ↔ ( 3 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 3 ≤ 3 ) ) |
33 |
4 4 31 32
|
mpbir3an |
⊢ 3 ∈ ( 0 ... 3 ) |
34 |
33 1
|
eleqtrri |
⊢ 3 ∈ 𝑉 |
35 |
1 2 3
|
konigsberglem3 |
⊢ ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) = 3 |
36 |
35
|
breq2i |
⊢ ( 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) ↔ 2 ∥ 3 ) |
37 |
8 36
|
mtbir |
⊢ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) |
38 |
|
fveq2 |
⊢ ( 𝑥 = 3 → ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) = ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) ) |
39 |
38
|
breq2d |
⊢ ( 𝑥 = 3 → ( 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) ↔ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) ) ) |
40 |
39
|
notbid |
⊢ ( 𝑥 = 3 → ( ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) ↔ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) ) ) |
41 |
40
|
elrab |
⊢ ( 3 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ↔ ( 3 ∈ 𝑉 ∧ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 3 ) ) ) |
42 |
34 37 41
|
mpbir2an |
⊢ 3 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } |
43 |
16 29 42
|
3pm3.2i |
⊢ ( 0 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ∧ 1 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ∧ 3 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ) |
44 |
|
c0ex |
⊢ 0 ∈ V |
45 |
|
1ex |
⊢ 1 ∈ V |
46 |
|
3ex |
⊢ 3 ∈ V |
47 |
44 45 46
|
tpss |
⊢ ( ( 0 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ∧ 1 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ∧ 3 ∈ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ) ↔ { 0 , 1 , 3 } ⊆ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } ) |
48 |
43 47
|
mpbi |
⊢ { 0 , 1 , 3 } ⊆ { 𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ( ( VtxDeg ‘ 𝐺 ) ‘ 𝑥 ) } |