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Theorem konigsberglem2 30189
Description: Lemma 2 for konigsberg 30193: Vertex 1 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
konigsberglem2 ((VtxDeg‘𝐺)‘1) = 3

Proof of Theorem konigsberglem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 7459 . . . 4 (0...3) ∈ V
2 s6cli 14895 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3484 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 28948 . . 3 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2735 . 2 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 1nn0 12542 . . 3 1 ∈ ℕ0
7 3nn0 12544 . . 3 3 ∈ ℕ0
8 1le3 12478 . . 3 1 ≤ 3
9 elfz2nn0 13648 . . 3 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
106, 7, 8, 9mpbir3an 1338 . 2 1 ∈ (0...3)
111, 3opiedgfvi 28949 . . 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}”⟩
1211eqcomi 2735 . 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}”⟩⟩)
13 s1cli 14615 . . 3 ⟨“{2, 3}”⟩ ∈ Word V
14 df-s7 14864 . . 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}”⟩)
15 eqid 2726 . . . 4 (0...3) = (0...3)
16 eqid 2726 . . . 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}”⟩
17 eqid 2726 . . . 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}”⟩⟩
1815, 16, 17konigsbergssiedgw 30186 . . 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})
192, 13, 14, 18mp3an 1458 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
20 s5cli 14894 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
2120elexi 3484 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
221, 21opvtxfvi 28948 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2322eqcomi 2735 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
241, 21opiedgfvi 28949 . . . 4 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
2524eqcomi 2735 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
26 s2cli 14891 . . . 4 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
27 s5s2 14946 . . . 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}”⟩)
2815, 16, 17konigsbergssiedgw 30186 . . . 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})
2920, 26, 27, 28mp3an 1458 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
30 s4cli 14893 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
3130elexi 3484 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
321, 31opvtxfvi 28948 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3332eqcomi 2735 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
341, 31opiedgfvi 28949 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3534eqcomi 2735 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
36 s3cli 14892 . . . . . 6 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
37 s4s3 14942 . . . . . 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}”⟩)
3815, 16, 17konigsbergssiedgw 30186 . . . . . 6 ((⟨“{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})
3930, 36, 37, 38mp3an 1458 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
40 s3cli 14892 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
4140elexi 3484 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
421, 41opvtxfvi 28948 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4342eqcomi 2735 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
441, 41opiedgfvi 28949 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4544eqcomi 2735 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
46 s4cli 14893 . . . . . . . 8 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
47 s3s4 14944 . . . . . . . 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}”⟩)
4815, 16, 17konigsbergssiedgw 30186 . . . . . . . 8 ((⟨“{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})
4940, 46, 47, 48mp3an 1458 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
50 s2cli 14891 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
5150elexi 3484 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ ∈ V
521, 51opvtxfvi 28948 . . . . . . . . 9 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5352eqcomi 2735 . . . . . . . 8 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
541, 51opiedgfvi 28949 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5554eqcomi 2735 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
56 s5cli 14894 . . . . . . . . 9 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
57 s2s5 14945 . . . . . . . . 9 ⟨“{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}”⟩)
5815, 16, 17konigsbergssiedgw 30186 . . . . . . . . 9 ((⟨“{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})
5950, 56, 57, 58mp3an 1458 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
60 s1cli 14615 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ Word V
6160elexi 3484 . . . . . . . . . . 11 ⟨“{0, 1}”⟩ ∈ V
621, 61opvtxfvi 28948 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6362eqcomi 2735 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
641, 61opiedgfvi 28949 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6564eqcomi 2735 . . . . . . . . 9 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
66 s6cli 14895 . . . . . . . . . 10 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
67 s1s6 14938 . . . . . . . . . 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}”⟩)
6815, 16, 17konigsbergssiedgw 30186 . . . . . . . . . 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})
6960, 66, 67, 68mp3an 1458 . . . . . . . . 9 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
70 0ex 5314 . . . . . . . . . . . . 13 ∅ ∈ V
711, 70opvtxfvi 28948 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7271eqcomi 2735 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
731, 70opiedgfvi 28949 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7473eqcomi 2735 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
75 wrd0 14549 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
76 eqid 2726 . . . . . . . . . . . 12 ∅ = ∅
7772, 74vtxdg0e 29414 . . . . . . . . . . . 12 ((1 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0)
7810, 76, 77mp2an 690 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0
79 0elfz 13654 . . . . . . . . . . . 12 (3 ∈ ℕ0 → 0 ∈ (0...3))
807, 79ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
81 0ne1 12337 . . . . . . . . . . 11 0 ≠ 1
82 s0s1 14933 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8364, 82eqtri 2754 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
8472, 10, 74, 75, 78, 62, 80, 81, 83vdegp1ci 29478 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = (0 + 1)
85 0p1e1 12388 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2754 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = 1
87 2nn0 12543 . . . . . . . . . 10 2 ∈ ℕ0
88 2re 12340 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 12346 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 12438 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 11389 . . . . . . . . . 10 2 ≤ 3
92 elfz2nn0 13648 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9387, 7, 91, 92mpbir3an 1338 . . . . . . . . 9 2 ∈ (0...3)
94 1ne2 12474 . . . . . . . . . 10 1 ≠ 2
9594necomi 2985 . . . . . . . . 9 2 ≠ 1
96 df-s2 14859 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9754, 96eqtri 2754 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9863, 10, 65, 69, 86, 52, 80, 81, 93, 95, 97vdegp1ai 29476 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘1) = 1
99 nn0fz0 13655 . . . . . . . . 9 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
1007, 99mpbi 229 . . . . . . . 8 3 ∈ (0...3)
101 1re 11266 . . . . . . . . 9 1 ∈ ℝ
102 1lt3 12439 . . . . . . . . 9 1 < 3
103101, 102gtneii 11378 . . . . . . . 8 3 ≠ 1
104 df-s3 14860 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10544, 104eqtri 2754 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10653, 10, 55, 59, 98, 42, 80, 81, 100, 103, 105vdegp1ai 29476 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘1) = 1
107 df-s4 14861 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10834, 107eqtri 2754 . . . . . . 7 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10943, 10, 45, 49, 106, 32, 93, 95, 108vdegp1bi 29477 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = (1 + 1)
110 1p1e2 12391 . . . . . 6 (1 + 1) = 2
111109, 110eqtri 2754 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = 2
112 df-s5 14862 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11324, 112eqtri 2754 . . . . 5 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11433, 10, 35, 39, 111, 22, 93, 95, 113vdegp1bi 29477 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = (2 + 1)
115 2p1e3 12408 . . . 4 (2 + 1) = 3
116114, 115eqtri 2754 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = 3
117 df-s6 14863 . . . 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}”⟩)
11811, 117eqtri 2754 . . 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}”⟩)
11923, 10, 25, 29, 116, 4, 93, 95, 100, 103, 118vdegp1ai 29476 . 2 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘1) = 3
120 konigsberg.v . . 3 𝑉 = (0...3)
121 konigsberg.e . . 3 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
122 konigsberg.g . . 3 𝐺 = ⟨𝑉, 𝐸
123120, 121, 122konigsbergvtx 30182 . 2 (Vtx‘𝐺) = (0...3)
124120, 121, 122konigsbergiedg 30183 . . 3 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
125124, 14eqtri 2754 . 2 (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
1265, 10, 12, 19, 119, 123, 93, 95, 100, 103, 125vdegp1ai 29476 1 ((VtxDeg‘𝐺)‘1) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cdif 3944  c0 4325  𝒫 cpw 4607  {csn 4633  {cpr 4635  cop 4639   class class class wbr 5155  cfv 6556  (class class class)co 7426  0cc0 11160  1c1 11161   + caddc 11163  cle 11301  2c2 12321  3c3 12322  0cn0 12526  ...cfz 13540  chash 14349  Word cword 14524   ++ cconcat 14580  ⟨“cs1 14605  ⟨“cs2 14852  ⟨“cs3 14853  ⟨“cs4 14854  ⟨“cs5 14855  ⟨“cs6 14856  ⟨“cs7 14857  Vtxcvtx 28935  iEdgciedg 28936  VtxDegcvtxdg 29405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8005  df-2nd 8006  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-1o 8498  df-2o 8499  df-oadd 8502  df-er 8736  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-dju 9946  df-card 9984  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12599  df-z 12613  df-uz 12877  df-xadd 13149  df-fz 13541  df-fzo 13684  df-hash 14350  df-word 14525  df-concat 14581  df-s1 14606  df-s2 14859  df-s3 14860  df-s4 14861  df-s5 14862  df-s6 14863  df-s7 14864  df-vtx 28937  df-iedg 28938  df-vtxdg 29406
This theorem is referenced by:  konigsberglem4  30191
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