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Theorem konigsberglem2 27775
Description: Lemma 2 for konigsberg 27779: 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 7002 . . . 4 (0...3) ∈ V
2 s6cli 14098 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3428 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 26487 . . 3 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2781 . 2 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 1nn0 11718 . . 3 1 ∈ ℕ0
7 3nn0 11720 . . 3 3 ∈ ℕ0
8 1le3 11652 . . 3 1 ≤ 3
9 elfz2nn0 12807 . . 3 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
106, 7, 8, 9mpbir3an 1321 . 2 1 ∈ (0...3)
111, 3opiedgfvi 26488 . . 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 2781 . 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 13758 . . 3 ⟨“{2, 3}”⟩ ∈ Word V
14 df-s7 14067 . . 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 2772 . . . 4 (0...3) = (0...3)
16 eqid 2772 . . . 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 2772 . . . 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 27772 . . 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 1440 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
20 s5cli 14097 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
2120elexi 3428 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
221, 21opvtxfvi 26487 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2322eqcomi 2781 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
241, 21opiedgfvi 26488 . . . 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 2781 . . 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 14094 . . . 4 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
27 s5s2 14149 . . . 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 27772 . . . 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 1440 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
30 s4cli 14096 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
3130elexi 3428 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
321, 31opvtxfvi 26487 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3332eqcomi 2781 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
341, 31opiedgfvi 26488 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3534eqcomi 2781 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
36 s3cli 14095 . . . . . 6 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
37 s4s3 14145 . . . . . 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 27772 . . . . . 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 1440 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
40 s3cli 14095 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
4140elexi 3428 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
421, 41opvtxfvi 26487 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4342eqcomi 2781 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
441, 41opiedgfvi 26488 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4544eqcomi 2781 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
46 s4cli 14096 . . . . . . . 8 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
47 s3s4 14147 . . . . . . . 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 27772 . . . . . . . 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 1440 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
50 s2cli 14094 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
5150elexi 3428 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ ∈ V
521, 51opvtxfvi 26487 . . . . . . . . 9 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5352eqcomi 2781 . . . . . . . 8 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
541, 51opiedgfvi 26488 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5554eqcomi 2781 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
56 s5cli 14097 . . . . . . . . 9 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
57 s2s5 14148 . . . . . . . . 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 27772 . . . . . . . . 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 1440 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
60 s1cli 13758 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ Word V
6160elexi 3428 . . . . . . . . . . 11 ⟨“{0, 1}”⟩ ∈ V
621, 61opvtxfvi 26487 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6362eqcomi 2781 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
641, 61opiedgfvi 26488 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6564eqcomi 2781 . . . . . . . . 9 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
66 s6cli 14098 . . . . . . . . . 10 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
67 s1s6 14141 . . . . . . . . . 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 27772 . . . . . . . . . 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 1440 . . . . . . . . 9 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
70 0ex 5062 . . . . . . . . . . . . 13 ∅ ∈ V
711, 70opvtxfvi 26487 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7271eqcomi 2781 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
731, 70opiedgfvi 26488 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7473eqcomi 2781 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
75 wrd0 13690 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
76 eqid 2772 . . . . . . . . . . . 12 ∅ = ∅
7772, 74vtxdg0e 26949 . . . . . . . . . . . 12 ((1 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0)
7810, 76, 77mp2an 679 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0
79 0elfz 12813 . . . . . . . . . . . 12 (3 ∈ ℕ0 → 0 ∈ (0...3))
807, 79ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
81 0ne1 11504 . . . . . . . . . . 11 0 ≠ 1
82 s0s1 14136 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8364, 82eqtri 2796 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
8472, 10, 74, 75, 78, 62, 80, 81, 83vdegp1ci 27013 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = (0 + 1)
85 0p1e1 11562 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2796 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = 1
87 2nn0 11719 . . . . . . . . . 10 2 ∈ ℕ0
88 2re 11507 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 11513 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 11612 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 10555 . . . . . . . . . 10 2 ≤ 3
92 elfz2nn0 12807 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9387, 7, 91, 92mpbir3an 1321 . . . . . . . . 9 2 ∈ (0...3)
94 1ne2 11648 . . . . . . . . . 10 1 ≠ 2
9594necomi 3015 . . . . . . . . 9 2 ≠ 1
96 df-s2 14062 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9754, 96eqtri 2796 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9863, 10, 65, 69, 86, 52, 80, 81, 93, 95, 97vdegp1ai 27011 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘1) = 1
99 nn0fz0 12814 . . . . . . . . 9 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
1007, 99mpbi 222 . . . . . . . 8 3 ∈ (0...3)
101 1re 10431 . . . . . . . . 9 1 ∈ ℝ
102 1lt3 11613 . . . . . . . . 9 1 < 3
103101, 102gtneii 10544 . . . . . . . 8 3 ≠ 1
104 df-s3 14063 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10544, 104eqtri 2796 . . . . . . . 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 27011 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘1) = 1
107 df-s4 14064 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10834, 107eqtri 2796 . . . . . . 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 27012 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = (1 + 1)
110 1p1e2 11565 . . . . . 6 (1 + 1) = 2
111109, 110eqtri 2796 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = 2
112 df-s5 14065 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11324, 112eqtri 2796 . . . . 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 27012 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = (2 + 1)
115 2p1e3 11582 . . . 4 (2 + 1) = 3
116114, 115eqtri 2796 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = 3
117 df-s6 14066 . . . 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 2796 . . 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 27011 . 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 27768 . 2 (Vtx‘𝐺) = (0...3)
124120, 121, 122konigsbergiedg 27769 . . 3 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
125124, 14eqtri 2796 . 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 27011 1 ((VtxDeg‘𝐺)‘1) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1507  wcel 2048  {crab 3086  Vcvv 3409  cdif 3822  c0 4173  𝒫 cpw 4416  {csn 4435  {cpr 4437  cop 4441   class class class wbr 4923  cfv 6182  (class class class)co 6970  0cc0 10327  1c1 10328   + caddc 10330  cle 10467  2c2 11488  3c3 11489  0cn0 11700  ...cfz 12701  chash 13498  Word cword 13662   ++ cconcat 13723  ⟨“cs1 13748  ⟨“cs2 14055  ⟨“cs3 14056  ⟨“cs4 14057  ⟨“cs5 14058  ⟨“cs6 14059  ⟨“cs7 14060  Vtxcvtx 26474  iEdgciedg 26475  VtxDegcvtxdg 26940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-dju 9116  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-3 11497  df-n0 11701  df-xnn0 11773  df-z 11787  df-uz 12052  df-xadd 12318  df-fz 12702  df-fzo 12843  df-hash 13499  df-word 13663  df-concat 13724  df-s1 13749  df-s2 14062  df-s3 14063  df-s4 14064  df-s5 14065  df-s6 14066  df-s7 14067  df-vtx 26476  df-iedg 26477  df-vtxdg 26941
This theorem is referenced by:  konigsberglem4  27777
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