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Theorem konigsberglem3 30282
Description: Lemma 3 for konigsberg 30285: Vertex 3 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
konigsberglem3 ((VtxDeg‘𝐺)‘3) = 3

Proof of Theorem konigsberglem3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 7463 . . . . 5 (0...3) ∈ V
2 s6cli 14919 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3500 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 29040 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2743 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 3nn0 12541 . . . 4 3 ∈ ℕ0
7 nn0fz0 13661 . . . 4 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
86, 7mpbi 230 . . 3 3 ∈ (0...3)
91, 3opiedgfvi 29041 . . . 4 (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 2743 . . 3 ⟨“{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 14639 . . . 4 ⟨“{2, 3}”⟩ ∈ Word V
12 df-s7 14888 . . . 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}”⟩)
13 eqid 2734 . . . . 5 (0...3) = (0...3)
14 eqid 2734 . . . . 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}”⟩
15 eqid 2734 . . . . 5 ⟨(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 30278 . . . 4 ((⟨“{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 1460 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
18 s5cli 14918 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
1918elexi 3500 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
201, 19opvtxfvi 29040 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2120eqcomi 2743 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
221, 19opiedgfvi 29041 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
2322eqcomi 2743 . . . . 5 ⟨“{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 14915 . . . . . 6 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25 s5s2 14970 . . . . . 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}”⟩)
2613, 14, 15konigsbergssiedgw 30278 . . . . . 6 ((⟨“{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 1460 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28 s4cli 14917 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
2928elexi 3500 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
301, 29opvtxfvi 29040 . . . . . . 7 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3130eqcomi 2743 . . . . . 6 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
321, 29opiedgfvi 29041 . . . . . . 7 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3332eqcomi 2743 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34 s3cli 14916 . . . . . . 7 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35 s4s3 14966 . . . . . . 7 ⟨“{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 30278 . . . . . . 7 ((⟨“{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 1460 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
38 s3cli 14916 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
3938elexi 3500 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
401, 39opvtxfvi 29040 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4140eqcomi 2743 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
421, 39opiedgfvi 29041 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4342eqcomi 2743 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44 s4cli 14917 . . . . . . . 8 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45 s3s4 14968 . . . . . . . 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}”⟩)
4613, 14, 15konigsbergssiedgw 30278 . . . . . . . 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})
4738, 44, 45, 46mp3an 1460 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
48 s2cli 14915 . . . . . . . . . . . 12 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
4948elexi 3500 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ ∈ V
501, 49opvtxfvi 29040 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5150eqcomi 2743 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
521, 49opiedgfvi 29041 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5352eqcomi 2743 . . . . . . . . 9 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54 s5cli 14918 . . . . . . . . . 10 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55 s2s5 14969 . . . . . . . . . 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}”⟩)
5613, 14, 15konigsbergssiedgw 30278 . . . . . . . . . 10 ((⟨“{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 1460 . . . . . . . . 9 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
58 s1cli 14639 . . . . . . . . . . . . 13 ⟨“{0, 1}”⟩ ∈ Word V
5958elexi 3500 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ V
601, 59opvtxfvi 29040 . . . . . . . . . . 11 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6160eqcomi 2743 . . . . . . . . . 10 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
621, 59opiedgfvi 29041 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6362eqcomi 2743 . . . . . . . . . 10 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64 s6cli 14919 . . . . . . . . . . 11 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65 s1s6 14962 . . . . . . . . . . 11 ⟨“{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 30278 . . . . . . . . . . 11 ((⟨“{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 1460 . . . . . . . . . 10 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
68 0ex 5312 . . . . . . . . . . . . 13 ∅ ∈ V
691, 68opvtxfvi 29040 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7069eqcomi 2743 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
711, 68opiedgfvi 29041 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7271eqcomi 2743 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
73 wrd0 14573 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
74 eqid 2734 . . . . . . . . . . . 12 ∅ = ∅
7570, 72vtxdg0e 29506 . . . . . . . . . . . 12 ((3 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0)
768, 74, 75mp2an 692 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0
77 0elfz 13660 . . . . . . . . . . . 12 (3 ∈ ℕ0 → 0 ∈ (0...3))
786, 77ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
79 3ne0 12369 . . . . . . . . . . . 12 3 ≠ 0
8079necomi 2992 . . . . . . . . . . 11 0 ≠ 3
81 1nn0 12539 . . . . . . . . . . . 12 1 ∈ ℕ0
82 1le3 12475 . . . . . . . . . . . 12 1 ≤ 3
83 elfz2nn0 13654 . . . . . . . . . . . 12 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
8481, 6, 82, 83mpbir3an 1340 . . . . . . . . . . 11 1 ∈ (0...3)
85 1re 11258 . . . . . . . . . . . 12 1 ∈ ℝ
86 1lt3 12436 . . . . . . . . . . . 12 1 < 3
8785, 86ltneii 11371 . . . . . . . . . . 11 1 ≠ 3
88 s0s1 14957 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8962, 88eqtri 2762 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
9070, 8, 72, 73, 76, 60, 78, 80, 84, 87, 89vdegp1ai 29568 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘3) = 0
91 2nn0 12540 . . . . . . . . . . 11 2 ∈ ℕ0
92 2re 12337 . . . . . . . . . . . 12 2 ∈ ℝ
93 3re 12343 . . . . . . . . . . . 12 3 ∈ ℝ
94 2lt3 12435 . . . . . . . . . . . 12 2 < 3
9592, 93, 94ltleii 11381 . . . . . . . . . . 11 2 ≤ 3
96 elfz2nn0 13654 . . . . . . . . . . 11 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9791, 6, 95, 96mpbir3an 1340 . . . . . . . . . 10 2 ∈ (0...3)
9892, 94ltneii 11371 . . . . . . . . . 10 2 ≠ 3
99 df-s2 14883 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
10052, 99eqtri 2762 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
10161, 8, 63, 67, 90, 50, 78, 80, 97, 98, 100vdegp1ai 29568 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘3) = 0
102 df-s3 14884 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10342, 102eqtri 2762 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10451, 8, 53, 57, 101, 40, 78, 80, 103vdegp1ci 29570 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = (0 + 1)
105 0p1e1 12385 . . . . . . . 8 (0 + 1) = 1
106104, 105eqtri 2762 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = 1
107 df-s4 14885 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10832, 107eqtri 2762 . . . . . . 7 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10941, 8, 43, 47, 106, 30, 84, 87, 97, 98, 108vdegp1ai 29568 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘3) = 1
110 df-s5 14886 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11122, 110eqtri 2762 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11231, 8, 33, 37, 109, 20, 84, 87, 97, 98, 111vdegp1ai 29568 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘3) = 1
113 df-s6 14887 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
1149, 113eqtri 2762 . . . . 5 (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}”⟩)
11521, 8, 23, 27, 112, 4, 97, 98, 114vdegp1ci 29570 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = (1 + 1)
116 1p1e2 12388 . . . 4 (1 + 1) = 2
117115, 116eqtri 2762 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = 2
118 konigsberg.v . . . 4 𝑉 = (0...3)
119 konigsberg.e . . . 4 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
120 konigsberg.g . . . 4 𝐺 = ⟨𝑉, 𝐸
121118, 119, 120konigsbergvtx 30274 . . 3 (Vtx‘𝐺) = (0...3)
122118, 119, 120konigsbergiedg 30275 . . . 4 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
123122, 12eqtri 2762 . . 3 (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
1245, 8, 10, 17, 117, 121, 97, 98, 123vdegp1ci 29570 . 2 ((VtxDeg‘𝐺)‘3) = (2 + 1)
125 2p1e3 12405 . 2 (2 + 1) = 3
126124, 125eqtri 2762 1 ((VtxDeg‘𝐺)‘3) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  cdif 3959  c0 4338  𝒫 cpw 4604  {csn 4630  {cpr 4632  cop 4636   class class class wbr 5147  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   + caddc 11155  cle 11293  2c2 12318  3c3 12319  0cn0 12523  ...cfz 13543  chash 14365  Word cword 14548   ++ cconcat 14604  ⟨“cs1 14629  ⟨“cs2 14876  ⟨“cs3 14877  ⟨“cs4 14878  ⟨“cs5 14879  ⟨“cs6 14880  ⟨“cs7 14881  Vtxcvtx 29027  iEdgciedg 29028  VtxDegcvtxdg 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-xadd 13152  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-s4 14885  df-s5 14886  df-s6 14887  df-s7 14888  df-vtx 29029  df-iedg 29030  df-vtxdg 29498
This theorem is referenced by:  konigsberglem4  30283
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