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Theorem konigsberglem2 27231
Description: Lemma 2 for konigsberg 27235: 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 6718 . . . 4 (0...3) ∈ V
2 s6cli 13675 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3244 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 25934 . . 3 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2660 . 2 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 1nn0 11346 . . 3 1 ∈ ℕ0
7 3nn0 11348 . . 3 3 ∈ ℕ0
8 1le3 11282 . . 3 1 ≤ 3
9 elfz2nn0 12469 . . 3 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
106, 7, 8, 9mpbir3an 1263 . 2 1 ∈ (0...3)
111, 3opiedgfvi 25935 . . 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 2660 . 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 13421 . . 3 ⟨“{2, 3}”⟩ ∈ Word V
14 df-s7 13644 . . 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 2651 . . . 4 (0...3) = (0...3)
16 eqid 2651 . . . 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 2651 . . . 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 27228 . . 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 1464 . 2 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
20 s5cli 13674 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
2120elexi 3244 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
221, 21opvtxfvi 25934 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2322eqcomi 2660 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
241, 21opiedgfvi 25935 . . . 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 2660 . . 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 13671 . . . 4 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
27 s5s2 13726 . . . 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 27228 . . . 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 1464 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
30 s4cli 13673 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
3130elexi 3244 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
321, 31opvtxfvi 25934 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3332eqcomi 2660 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
341, 31opiedgfvi 25935 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3534eqcomi 2660 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
36 s3cli 13672 . . . . . 6 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
37 s4s3 13722 . . . . . 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 27228 . . . . . 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 1464 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
40 s3cli 13672 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
4140elexi 3244 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
421, 41opvtxfvi 25934 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4342eqcomi 2660 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
441, 41opiedgfvi 25935 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4544eqcomi 2660 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
46 s4cli 13673 . . . . . . . 8 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
47 s3s4 13724 . . . . . . . 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 27228 . . . . . . . 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 1464 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
50 s2cli 13671 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
5150elexi 3244 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ ∈ V
521, 51opvtxfvi 25934 . . . . . . . . 9 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5352eqcomi 2660 . . . . . . . 8 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
541, 51opiedgfvi 25935 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5554eqcomi 2660 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
56 s5cli 13674 . . . . . . . . 9 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
57 s2s5 13725 . . . . . . . . 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 27228 . . . . . . . . 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 1464 . . . . . . . 8 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
60 s1cli 13421 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ Word V
6160elexi 3244 . . . . . . . . . . 11 ⟨“{0, 1}”⟩ ∈ V
621, 61opvtxfvi 25934 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6362eqcomi 2660 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
641, 61opiedgfvi 25935 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6564eqcomi 2660 . . . . . . . . 9 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
66 s6cli 13675 . . . . . . . . . 10 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
67 s1s6 13718 . . . . . . . . . 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 27228 . . . . . . . . . 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 1464 . . . . . . . . 9 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
70 0ex 4823 . . . . . . . . . . . . 13 ∅ ∈ V
711, 70opvtxfvi 25934 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7271eqcomi 2660 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
731, 70opiedgfvi 25935 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7473eqcomi 2660 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
75 wrd0 13362 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
76 eqid 2651 . . . . . . . . . . . 12 ∅ = ∅
7772, 74vtxdg0e 26426 . . . . . . . . . . . 12 ((1 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0)
7810, 76, 77mp2an 708 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0
79 0elfz 12475 . . . . . . . . . . . 12 (3 ∈ ℕ0 → 0 ∈ (0...3))
807, 79ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
81 0ne1 11126 . . . . . . . . . . 11 0 ≠ 1
82 s0s1 13713 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8364, 82eqtri 2673 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
8472, 10, 74, 75, 78, 62, 80, 81, 83vdegp1ci 26490 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = (0 + 1)
85 0p1e1 11170 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2673 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = 1
87 2nn0 11347 . . . . . . . . . 10 2 ∈ ℕ0
88 2re 11128 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 11132 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 11233 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 10198 . . . . . . . . . 10 2 ≤ 3
92 elfz2nn0 12469 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9387, 7, 91, 92mpbir3an 1263 . . . . . . . . 9 2 ∈ (0...3)
94 1ne2 11278 . . . . . . . . . 10 1 ≠ 2
9594necomi 2877 . . . . . . . . 9 2 ≠ 1
96 df-s2 13639 . . . . . . . . . 10 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9754, 96eqtri 2673 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
9863, 10, 65, 69, 86, 52, 80, 81, 93, 95, 97vdegp1ai 26488 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘1) = 1
99 nn0fz0 12476 . . . . . . . . 9 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
1007, 99mpbi 220 . . . . . . . 8 3 ∈ (0...3)
101 1re 10077 . . . . . . . . 9 1 ∈ ℝ
102 1lt3 11234 . . . . . . . . 9 1 < 3
103101, 102gtneii 10187 . . . . . . . 8 3 ≠ 1
104 df-s3 13640 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10544, 104eqtri 2673 . . . . . . . 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 26488 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘1) = 1
107 df-s4 13641 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10834, 107eqtri 2673 . . . . . . 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 26489 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = (1 + 1)
110 1p1e2 11172 . . . . . 6 (1 + 1) = 2
111109, 110eqtri 2673 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = 2
112 df-s5 13642 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11324, 112eqtri 2673 . . . . 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 26489 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = (2 + 1)
115 2p1e3 11189 . . . 4 (2 + 1) = 3
116114, 115eqtri 2673 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = 3
117 df-s6 13643 . . . 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 2673 . . 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 26488 . 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 27224 . 2 (Vtx‘𝐺) = (0...3)
124120, 121, 122konigsbergiedg 27225 . . 3 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
125124, 14eqtri 2673 . 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 26488 1 ((VtxDeg‘𝐺)‘1) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cdif 3604  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cle 10113  2c2 11108  3c3 11109  0cn0 11330  ...cfz 12364  #chash 13157  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326  ⟨“cs2 13632  ⟨“cs3 13633  ⟨“cs4 13634  ⟨“cs5 13635  ⟨“cs6 13636  ⟨“cs7 13637  Vtxcvtx 25919  iEdgciedg 25920  VtxDegcvtxdg 26417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-xadd 11985  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-s4 13641  df-s5 13642  df-s6 13643  df-s7 13644  df-vtx 25921  df-iedg 25922  df-vtxdg 26418
This theorem is referenced by:  konigsberglem4  27233
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