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Theorem konigsberglem2 27426
Description: Lemma 2 for konigsberg 27430: 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 6842 . . . 4 (0...3) ∈ V
2 s6cli 13849 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3353 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 26109 . . 3 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2769 . 2 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 1nn0 11520 . . 3 1 ∈ ℕ0
7 3nn0 11522 . . 3 3 ∈ ℕ0
8 1le3 11456 . . 3 1 ≤ 3
9 elfz2nn0 12644 . . 3 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
106, 7, 8, 9mpbir3an 1427 . 2 1 ∈ (0...3)
111, 3opiedgfvi 26110 . . 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 2769 . 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 13595 . . 3 ⟨“{2, 3}”⟩ ∈ Word V
14 df-s7 13818 . . 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 2760 . . . 4 (0...3) = (0...3)
16 eqid 2760 . . . 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 2760 . . . 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 27423 . . 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 1573 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
20 s5cli 13848 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
2120elexi 3353 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
221, 21opvtxfvi 26109 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2322eqcomi 2769 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
241, 21opiedgfvi 26110 . . . 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 2769 . . 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 13845 . . . 4 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
27 s5s2 13900 . . . 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 27423 . . . 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 1573 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
30 s4cli 13847 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
3130elexi 3353 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
321, 31opvtxfvi 26109 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3332eqcomi 2769 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
341, 31opiedgfvi 26110 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3534eqcomi 2769 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
36 s3cli 13846 . . . . . 6 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
37 s4s3 13896 . . . . . 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 27423 . . . . . 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 1573 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
40 s3cli 13846 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
4140elexi 3353 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
421, 41opvtxfvi 26109 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4342eqcomi 2769 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
441, 41opiedgfvi 26110 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4544eqcomi 2769 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
46 s4cli 13847 . . . . . . . 8 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
47 s3s4 13898 . . . . . . . 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 27423 . . . . . . . 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 1573 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
50 s2cli 13845 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
5150elexi 3353 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ ∈ V
521, 51opvtxfvi 26109 . . . . . . . . 9 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5352eqcomi 2769 . . . . . . . 8 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
541, 51opiedgfvi 26110 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5554eqcomi 2769 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
56 s5cli 13848 . . . . . . . . 9 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
57 s2s5 13899 . . . . . . . . 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 27423 . . . . . . . . 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 1573 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
60 s1cli 13595 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ Word V
6160elexi 3353 . . . . . . . . . . 11 ⟨“{0, 1}”⟩ ∈ V
621, 61opvtxfvi 26109 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6362eqcomi 2769 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
641, 61opiedgfvi 26110 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6564eqcomi 2769 . . . . . . . . 9 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
66 s6cli 13849 . . . . . . . . . 10 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
67 s1s6 13892 . . . . . . . . . 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 27423 . . . . . . . . . 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 1573 . . . . . . . . 9 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
70 0ex 4942 . . . . . . . . . . . . 13 ∅ ∈ V
711, 70opvtxfvi 26109 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7271eqcomi 2769 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
731, 70opiedgfvi 26110 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7473eqcomi 2769 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
75 wrd0 13536 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
76 eqid 2760 . . . . . . . . . . . 12 ∅ = ∅
7772, 74vtxdg0e 26601 . . . . . . . . . . . 12 ((1 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0)
7810, 76, 77mp2an 710 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0
79 0elfz 12650 . . . . . . . . . . . 12 (3 ∈ ℕ0 → 0 ∈ (0...3))
807, 79ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
81 0ne1 11300 . . . . . . . . . . 11 0 ≠ 1
82 s0s1 13887 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8364, 82eqtri 2782 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
8472, 10, 74, 75, 78, 62, 80, 81, 83vdegp1ci 26665 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = (0 + 1)
85 0p1e1 11344 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2782 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = 1
87 2nn0 11521 . . . . . . . . . 10 2 ∈ ℕ0
88 2re 11302 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 11306 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 11407 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 10372 . . . . . . . . . 10 2 ≤ 3
92 elfz2nn0 12644 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9387, 7, 91, 92mpbir3an 1427 . . . . . . . . 9 2 ∈ (0...3)
94 1ne2 11452 . . . . . . . . . 10 1 ≠ 2
9594necomi 2986 . . . . . . . . 9 2 ≠ 1
96 df-s2 13813 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9754, 96eqtri 2782 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9863, 10, 65, 69, 86, 52, 80, 81, 93, 95, 97vdegp1ai 26663 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘1) = 1
99 nn0fz0 12651 . . . . . . . . 9 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
1007, 99mpbi 220 . . . . . . . 8 3 ∈ (0...3)
101 1re 10251 . . . . . . . . 9 1 ∈ ℝ
102 1lt3 11408 . . . . . . . . 9 1 < 3
103101, 102gtneii 10361 . . . . . . . 8 3 ≠ 1
104 df-s3 13814 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10544, 104eqtri 2782 . . . . . . . 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 26663 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘1) = 1
107 df-s4 13815 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10834, 107eqtri 2782 . . . . . . 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 26664 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = (1 + 1)
110 1p1e2 11346 . . . . . 6 (1 + 1) = 2
111109, 110eqtri 2782 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = 2
112 df-s5 13816 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11324, 112eqtri 2782 . . . . 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 26664 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = (2 + 1)
115 2p1e3 11363 . . . 4 (2 + 1) = 3
116114, 115eqtri 2782 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = 3
117 df-s6 13817 . . . 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 2782 . . 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 26663 . 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 27419 . 2 (Vtx‘𝐺) = (0...3)
124120, 121, 122konigsbergiedg 27420 . . 3 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
125124, 14eqtri 2782 . 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 26663 1 ((VtxDeg‘𝐺)‘1) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cdif 3712  c0 4058  𝒫 cpw 4302  {csn 4321  {cpr 4323  cop 4327   class class class wbr 4804  cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149   + caddc 10151  cle 10287  2c2 11282  3c3 11283  0cn0 11504  ...cfz 12539  chash 13331  Word cword 13497   ++ cconcat 13499  ⟨“cs1 13500  ⟨“cs2 13806  ⟨“cs3 13807  ⟨“cs4 13808  ⟨“cs5 13809  ⟨“cs6 13810  ⟨“cs7 13811  Vtxcvtx 26094  iEdgciedg 26095  VtxDegcvtxdg 26592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-xadd 12160  df-fz 12540  df-fzo 12680  df-hash 13332  df-word 13505  df-concat 13507  df-s1 13508  df-s2 13813  df-s3 13814  df-s4 13815  df-s5 13816  df-s6 13817  df-s7 13818  df-vtx 26096  df-iedg 26097  df-vtxdg 26593
This theorem is referenced by:  konigsberglem4  27428
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