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Theorem konigsberglem1 27094
Description: Lemma 1 for konigsberg 27099: 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 6663 . . . 4 (0...3) ∈ V
2 s6cli 13610 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3208 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 25870 . . 3 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2629 . 2 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 3nn0 11295 . . 3 3 ∈ ℕ0
7 0elfz 12420 . . 3 (3 ∈ ℕ0 → 0 ∈ (0...3))
86, 7ax-mp 5 . 2 0 ∈ (0...3)
91, 3opiedgfvi 25871 . . 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 2629 . 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 13367 . . 3 ⟨“{2, 3}”⟩ ∈ Word V
12 df-s7 13579 . . 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 2620 . . . 4 (0...3) = (0...3)
14 eqid 2620 . . . 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 2620 . . . 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 27091 . . 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 1422 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
18 s5cli 13609 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
1918elexi 3208 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
201, 19opvtxfvi 25870 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2120eqcomi 2629 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
221, 19opiedgfvi 25871 . . . 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 2629 . . 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 13606 . . . 4 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25 s5s2 13661 . . . 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 27091 . . . 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 1422 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
28 s4cli 13608 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
2928elexi 3208 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
301, 29opvtxfvi 25870 . . . . 5 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3130eqcomi 2629 . . . 4 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
321, 29opiedgfvi 25871 . . . . 5 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3332eqcomi 2629 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34 s3cli 13607 . . . . 5 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35 s4s3 13657 . . . . 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 27091 . . . . 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 1422 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
38 s3cli 13607 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
3938elexi 3208 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
401, 39opvtxfvi 25870 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4140eqcomi 2629 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
421, 39opiedgfvi 25871 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4342eqcomi 2629 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44 s4cli 13608 . . . . . 6 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45 s3s4 13659 . . . . . 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 27091 . . . . . 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 1422 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
48 s2cli 13606 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
4948elexi 3208 . . . . . . . . 9 ⟨“{0, 1} {0, 2}”⟩ ∈ V
501, 49opvtxfvi 25870 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5150eqcomi 2629 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
521, 49opiedgfvi 25871 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5352eqcomi 2629 . . . . . . 7 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54 s5cli 13609 . . . . . . . 8 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55 s2s5 13660 . . . . . . . 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 27091 . . . . . . . 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 1422 . . . . . . 7 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
58 s1cli 13367 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ Word V
5958elexi 3208 . . . . . . . . . . 11 ⟨“{0, 1}”⟩ ∈ V
601, 59opvtxfvi 25870 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6160eqcomi 2629 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
621, 59opiedgfvi 25871 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6362eqcomi 2629 . . . . . . . . 9 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64 s6cli 13610 . . . . . . . . . 10 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65 s1s6 13653 . . . . . . . . . 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 27091 . . . . . . . . . 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 1422 . . . . . . . . 9 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
68 0ex 4781 . . . . . . . . . . . . 13 ∅ ∈ V
691, 68opvtxfvi 25870 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7069eqcomi 2629 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
711, 68opiedgfvi 25871 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7271eqcomi 2629 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
73 wrd0 13313 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
74 eqid 2620 . . . . . . . . . . . 12 ∅ = ∅
7570, 72vtxdg0e 26351 . . . . . . . . . . . 12 ((0 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0)
768, 74, 75mp2an 707 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0
77 1nn0 11293 . . . . . . . . . . . 12 1 ∈ ℕ0
78 1le3 11229 . . . . . . . . . . . 12 1 ≤ 3
79 elfz2nn0 12415 . . . . . . . . . . . 12 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
8077, 6, 78, 79mpbir3an 1242 . . . . . . . . . . 11 1 ∈ (0...3)
81 ax-1ne0 9990 . . . . . . . . . . 11 1 ≠ 0
82 s0s1 13648 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8362, 82eqtri 2642 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
8470, 8, 72, 73, 76, 60, 80, 81, 83vdegp1bi 26414 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = (0 + 1)
85 0p1e1 11117 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2642 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = 1
87 2nn0 11294 . . . . . . . . . 10 2 ∈ ℕ0
88 2re 11075 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 11079 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 11180 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 10145 . . . . . . . . . 10 2 ≤ 3
92 elfz2nn0 12415 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9387, 6, 91, 92mpbir3an 1242 . . . . . . . . 9 2 ∈ (0...3)
94 2ne0 11098 . . . . . . . . 9 2 ≠ 0
95 df-s2 13574 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9652, 95eqtri 2642 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9761, 8, 63, 67, 86, 50, 93, 94, 96vdegp1bi 26414 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = (1 + 1)
98 1p1e2 11119 . . . . . . . 8 (1 + 1) = 2
9997, 98eqtri 2642 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = 2
100 nn0fz0 12421 . . . . . . . 8 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
1016, 100mpbi 220 . . . . . . 7 3 ∈ (0...3)
102 3ne0 11100 . . . . . . 7 3 ≠ 0
103 df-s3 13575 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10442, 103eqtri 2642 . . . . . . 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 26414 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = (2 + 1)
106 2p1e3 11136 . . . . . 6 (2 + 1) = 3
107105, 106eqtri 2642 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = 3
108 df-s4 13576 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10932, 108eqtri 2642 . . . . 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 26413 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘0) = 3
111 df-s5 13577 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11222, 111eqtri 2642 . . . 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 26413 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘0) = 3
114 df-s6 13578 . . . 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 2642 . . 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 26413 . 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 27086 . 2 (Vtx‘𝐺) = (0...3)
121117, 118, 119konigsbergiedg 27087 . . 3 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
122121, 12eqtri 2642 . 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 26413 1 ((VtxDeg‘𝐺)‘0) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cdif 3564  c0 3907  𝒫 cpw 4149  {csn 4168  {cpr 4170  cop 4174   class class class wbr 4644  cfv 5876  (class class class)co 6635  0cc0 9921  1c1 9922   + caddc 9924  cle 10060  2c2 11055  3c3 11056  0cn0 11277  ...cfz 12311  #chash 13100  Word cword 13274   ++ cconcat 13276  ⟨“cs1 13277  ⟨“cs2 13567  ⟨“cs3 13568  ⟨“cs4 13569  ⟨“cs5 13570  ⟨“cs6 13571  ⟨“cs7 13572  Vtxcvtx 25855  iEdgciedg 25856  VtxDegcvtxdg 26342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-xnn0 11349  df-z 11363  df-uz 11673  df-xadd 11932  df-fz 12312  df-fzo 12450  df-hash 13101  df-word 13282  df-concat 13284  df-s1 13285  df-s2 13574  df-s3 13575  df-s4 13576  df-s5 13577  df-s6 13578  df-s7 13579  df-vtx 25857  df-iedg 25858  df-vtxdg 26343
This theorem is referenced by:  konigsberglem4  27097
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