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Theorem konigsberglem1 27488
Description: Lemma 1 for konigsberg 27493: Vertex 0 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
Hypotheses
Ref Expression
konigsberg.v 𝑉 = (0...3)
konigsberg.e 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
konigsberg.g 𝐺 = ⟨𝑉, 𝐸
Assertion
Ref Expression
konigsberglem1 ((VtxDeg‘𝐺)‘0) = 3

Proof of Theorem konigsberglem1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 6874 . . . 4 (0...3) ∈ V
2 s6cli 13915 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3366 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 26178 . . 3 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2774 . 2 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 3nn0 11558 . . 3 3 ∈ ℕ0
7 0elfz 12644 . . 3 (3 ∈ ℕ0 → 0 ∈ (0...3))
86, 7ax-mp 5 . 2 0 ∈ (0...3)
91, 3opiedgfvi 26179 . . 3 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩
109eqcomi 2774 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
11 s1cli 13576 . . 3 ⟨“{2, 3}”⟩ ∈ Word V
12 df-s7 13884 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
13 eqid 2765 . . . 4 (0...3) = (0...3)
14 eqid 2765 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
15 eqid 2765 . . . 4 ⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩⟩ = ⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩⟩
1613, 14, 15konigsbergssiedgw 27486 . . 3 ((⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V ∧ ⟨“{2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
172, 11, 12, 16mp3an 1585 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
18 s5cli 13914 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
1918elexi 3366 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
201, 19opvtxfvi 26178 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2120eqcomi 2774 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
221, 19opiedgfvi 26179 . . . 4 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
2322eqcomi 2774 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
24 s2cli 13911 . . . 4 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25 s5s2 13966 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3} {2, 3}”⟩)
2613, 14, 15konigsbergssiedgw 27486 . . . 4 ((⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V ∧ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
2718, 24, 25, 26mp3an 1585 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28 s4cli 13913 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
2928elexi 3366 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
301, 29opvtxfvi 26178 . . . . 5 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3130eqcomi 2774 . . . 4 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
321, 29opiedgfvi 26179 . . . . 5 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3332eqcomi 2774 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34 s3cli 13912 . . . . 5 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35 s4s3 13962 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2} {2, 3} {2, 3}”⟩)
3613, 14, 15konigsbergssiedgw 27486 . . . . 5 ((⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V ∧ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
3728, 34, 35, 36mp3an 1585 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
38 s3cli 13912 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
3938elexi 3366 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
401, 39opvtxfvi 26178 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4140eqcomi 2774 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
421, 39opiedgfvi 26179 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4342eqcomi 2774 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44 s4cli 13913 . . . . . 6 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45 s3s4 13964 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩)
4613, 14, 15konigsbergssiedgw 27486 . . . . . 6 ((⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V ∧ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4738, 44, 45, 46mp3an 1585 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
48 s2cli 13911 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
4948elexi 3366 . . . . . . . . 9 ⟨“{0, 1} {0, 2}”⟩ ∈ V
501, 49opvtxfvi 26178 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5150eqcomi 2774 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
521, 49opiedgfvi 26179 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5352eqcomi 2774 . . . . . . 7 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54 s5cli 13914 . . . . . . . 8 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55 s2s5 13965 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)
5613, 14, 15konigsbergssiedgw 27486 . . . . . . . 8 ((⟨“{0, 1} {0, 2}”⟩ ∈ Word V ∧ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
5748, 54, 55, 56mp3an 1585 . . . . . . 7 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
58 s1cli 13576 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ Word V
5958elexi 3366 . . . . . . . . . . 11 ⟨“{0, 1}”⟩ ∈ V
601, 59opvtxfvi 26178 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6160eqcomi 2774 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
621, 59opiedgfvi 26179 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6362eqcomi 2774 . . . . . . . . 9 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64 s6cli 13915 . . . . . . . . . 10 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65 s1s6 13958 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)
6613, 14, 15konigsbergssiedgw 27486 . . . . . . . . . 10 ((⟨“{0, 1}”⟩ ∈ Word V ∧ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6758, 64, 65, 66mp3an 1585 . . . . . . . . 9 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
68 0ex 4950 . . . . . . . . . . . . 13 ∅ ∈ V
691, 68opvtxfvi 26178 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7069eqcomi 2774 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
711, 68opiedgfvi 26179 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7271eqcomi 2774 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
73 wrd0 13511 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
74 eqid 2765 . . . . . . . . . . . 12 ∅ = ∅
7570, 72vtxdg0e 26661 . . . . . . . . . . . 12 ((0 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0)
768, 74, 75mp2an 683 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0
77 1nn0 11556 . . . . . . . . . . . 12 1 ∈ ℕ0
78 1le3 11490 . . . . . . . . . . . 12 1 ≤ 3
79 elfz2nn0 12638 . . . . . . . . . . . 12 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
8077, 6, 78, 79mpbir3an 1441 . . . . . . . . . . 11 1 ∈ (0...3)
81 ax-1ne0 10258 . . . . . . . . . . 11 1 ≠ 0
82 s0s1 13953 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8362, 82eqtri 2787 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
8470, 8, 72, 73, 76, 60, 80, 81, 83vdegp1bi 26724 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = (0 + 1)
85 0p1e1 11401 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2787 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = 1
87 2nn0 11557 . . . . . . . . . 10 2 ∈ ℕ0
88 2re 11346 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 11352 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 11450 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 10414 . . . . . . . . . 10 2 ≤ 3
92 elfz2nn0 12638 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9387, 6, 91, 92mpbir3an 1441 . . . . . . . . 9 2 ∈ (0...3)
94 2ne0 11383 . . . . . . . . 9 2 ≠ 0
95 df-s2 13879 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9652, 95eqtri 2787 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9761, 8, 63, 67, 86, 50, 93, 94, 96vdegp1bi 26724 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = (1 + 1)
98 1p1e2 11404 . . . . . . . 8 (1 + 1) = 2
9997, 98eqtri 2787 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = 2
100 nn0fz0 12645 . . . . . . . 8 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
1016, 100mpbi 221 . . . . . . 7 3 ∈ (0...3)
102 3ne0 11385 . . . . . . 7 3 ≠ 0
103 df-s3 13880 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10442, 103eqtri 2787 . . . . . . 7 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10551, 8, 53, 57, 99, 40, 101, 102, 104vdegp1bi 26724 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = (2 + 1)
106 2p1e3 11421 . . . . . 6 (2 + 1) = 3
107105, 106eqtri 2787 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = 3
108 df-s4 13881 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10932, 108eqtri 2787 . . . . 5 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
11041, 8, 43, 47, 107, 30, 80, 81, 93, 94, 109vdegp1ai 26723 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘0) = 3
111 df-s5 13882 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11222, 111eqtri 2787 . . . 4 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11331, 8, 33, 37, 110, 20, 80, 81, 93, 94, 112vdegp1ai 26723 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘0) = 3
114 df-s6 13883 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
1159, 114eqtri 2787 . . 3 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
11621, 8, 23, 27, 113, 4, 93, 94, 101, 102, 115vdegp1ai 26723 . 2 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘0) = 3
117 konigsberg.v . . 3 𝑉 = (0...3)
118 konigsberg.e . . 3 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
119 konigsberg.g . . 3 𝐺 = ⟨𝑉, 𝐸
120117, 118, 119konigsbergvtx 27482 . 2 (Vtx‘𝐺) = (0...3)
121117, 118, 119konigsbergiedg 27483 . . 3 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
122121, 12eqtri 2787 . 2 (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
1235, 8, 10, 17, 116, 120, 93, 94, 101, 102, 122vdegp1ai 26723 1 ((VtxDeg‘𝐺)‘0) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  cdif 3729  c0 4079  𝒫 cpw 4315  {csn 4334  {cpr 4336  cop 4340   class class class wbr 4809  cfv 6068  (class class class)co 6842  0cc0 10189  1c1 10190   + caddc 10192  cle 10329  2c2 11327  3c3 11328  0cn0 11538  ...cfz 12533  chash 13321  Word cword 13486   ++ cconcat 13541  ⟨“cs1 13566  ⟨“cs2 13872  ⟨“cs3 13873  ⟨“cs4 13874  ⟨“cs5 13875  ⟨“cs6 13876  ⟨“cs7 13877  Vtxcvtx 26165  iEdgciedg 26166  VtxDegcvtxdg 26652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-xadd 12147  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-concat 13542  df-s1 13567  df-s2 13879  df-s3 13880  df-s4 13881  df-s5 13882  df-s6 13883  df-s7 13884  df-vtx 26167  df-iedg 26168  df-vtxdg 26653
This theorem is referenced by:  konigsberglem4  27491
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