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Theorem konigsberglem1 30271

Description: Lemma 1 for konigsberg 30276: Vertex 0 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
konigsberglem1	⊢ ((VtxDeg‘𝐺)‘0) = 3

Proof of Theorem konigsberglem1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7464	. . . 4 ⊢ (0...3) ∈ V
2		s6cli 14923	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
3	2	elexi 3503	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
4	1, 3	opvtxfvi 29026	. . 3 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
5	4	eqcomi 2746	. 2 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6		3nn0 12544	. . 3 ⊢ 3 ∈ ℕ₀
7		0elfz 13664	. . 3 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
8	6, 7	ax-mp 5	. 2 ⊢ 0 ∈ (0...3)
9	1, 3	opiedgfvi 29027	. . 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}”⟩
10	9	eqcomi 2746	. 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}”⟩⟩)
11		s1cli 14643	. . 3 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
12		df-s7 14892	. . 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}”⟩)
13		eqid 2737	. . . 4 ⊢ (0...3) = (0...3)
14		eqid 2737	. . . 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}”⟩
15		eqid 2737	. . . 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}”⟩⟩
16	13, 14, 15	konigsbergssiedgw 30269	. . 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})
17	2, 11, 12, 16	mp3an 1463	. 2 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
18		s5cli 14922	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
19	18	elexi 3503	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
20	1, 19	opvtxfvi 29026	. . . 4 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
21	20	eqcomi 2746	. . 3 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
22	1, 19	opiedgfvi 29027	. . . 4 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
23	22	eqcomi 2746	. . 3 ⊢ ⟨“{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 14919	. . . 4 ⊢ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25		s5s2 14974	. . . 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}”⟩)
26	13, 14, 15	konigsbergssiedgw 30269	. . . 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})
27	18, 24, 25, 26	mp3an 1463	. . 3 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28		s4cli 14921	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
29	28	elexi 3503	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
30	1, 29	opvtxfvi 29026	. . . . 5 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
31	30	eqcomi 2746	. . . 4 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
32	1, 29	opiedgfvi 29027	. . . . 5 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
33	32	eqcomi 2746	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34		s3cli 14920	. . . . 5 ⊢ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35		s4s3 14970	. . . . 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}”⟩)
36	13, 14, 15	konigsbergssiedgw 30269	. . . . 5 ⊢ ((⟨“{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})
37	28, 34, 35, 36	mp3an 1463	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
38		s3cli 14920	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
39	38	elexi 3503	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
40	1, 39	opvtxfvi 29026	. . . . . 6 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
41	40	eqcomi 2746	. . . . 5 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
42	1, 39	opiedgfvi 29027	. . . . . 6 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
43	42	eqcomi 2746	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44		s4cli 14921	. . . . . 6 ⊢ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45		s3s4 14972	. . . . . 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}”⟩)
46	13, 14, 15	konigsbergssiedgw 30269	. . . . . 6 ⊢ ((⟨“{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})
47	38, 44, 45, 46	mp3an 1463	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
48		s2cli 14919	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
49	48	elexi 3503	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
50	1, 49	opvtxfvi 29026	. . . . . . . 8 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
51	50	eqcomi 2746	. . . . . . 7 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
52	1, 49	opiedgfvi 29027	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
53	52	eqcomi 2746	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54		s5cli 14922	. . . . . . . 8 ⊢ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55		s2s5 14973	. . . . . . . 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}”⟩)
56	13, 14, 15	konigsbergssiedgw 30269	. . . . . . . 8 ⊢ ((⟨“{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})
57	48, 54, 55, 56	mp3an 1463	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
58		s1cli 14643	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
59	58	elexi 3503	. . . . . . . . . . 11 ⊢ ⟨“{0, 1}”⟩ ∈ V
60	1, 59	opvtxfvi 29026	. . . . . . . . . 10 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
61	60	eqcomi 2746	. . . . . . . . 9 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
62	1, 59	opiedgfvi 29027	. . . . . . . . . 10 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
63	62	eqcomi 2746	. . . . . . . . 9 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64		s6cli 14923	. . . . . . . . . 10 ⊢ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65		s1s6 14966	. . . . . . . . . 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}”⟩)
66	13, 14, 15	konigsbergssiedgw 30269	. . . . . . . . . 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})
67	58, 64, 65, 66	mp3an 1463	. . . . . . . . 9 ⊢ ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
68		0ex 5307	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
69	1, 68	opvtxfvi 29026	. . . . . . . . . . . 12 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
70	69	eqcomi 2746	. . . . . . . . . . 11 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
71	1, 68	opiedgfvi 29027	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
72	71	eqcomi 2746	. . . . . . . . . . 11 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
73		wrd0 14577	. . . . . . . . . . 11 ⊢ ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
74		eqid 2737	. . . . . . . . . . . 12 ⊢ ∅ = ∅
75	70, 72	vtxdg0e 29492	. . . . . . . . . . . 12 ⊢ ((0 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0)
76	8, 74, 75	mp2an 692	. . . . . . . . . . 11 ⊢ ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0
77		1nn0 12542	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
78		1le3 12478	. . . . . . . . . . . 12 ⊢ 1 ≤ 3
79		elfz2nn0 13658	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
80	77, 6, 78, 79	mpbir3an 1342	. . . . . . . . . . 11 ⊢ 1 ∈ (0...3)
81		ax-1ne0 11224	. . . . . . . . . . 11 ⊢ 1 ≠ 0
82		s0s1 14961	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
83	62, 82	eqtri 2765	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
84	70, 8, 72, 73, 76, 60, 80, 81, 83	vdegp1bi 29555	. . . . . . . . . 10 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = (0 + 1)
85		0p1e1 12388	. . . . . . . . . 10 ⊢ (0 + 1) = 1
86	84, 85	eqtri 2765	. . . . . . . . 9 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = 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
91	88, 89, 90	ltleii 11384	. . . . . . . . . 10 ⊢ 2 ≤ 3
92		elfz2nn0 13658	. . . . . . . . . 10 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
93	87, 6, 91, 92	mpbir3an 1342	. . . . . . . . 9 ⊢ 2 ∈ (0...3)
94		2ne0 12370	. . . . . . . . 9 ⊢ 2 ≠ 0
95		df-s2 14887	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
96	52, 95	eqtri 2765	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
97	61, 8, 63, 67, 86, 50, 93, 94, 96	vdegp1bi 29555	. . . . . . . 8 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = (1 + 1)
98		1p1e2 12391	. . . . . . . 8 ⊢ (1 + 1) = 2
99	97, 98	eqtri 2765	. . . . . . 7 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = 2
100		nn0fz0 13665	. . . . . . . 8 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
101	6, 100	mpbi 230	. . . . . . 7 ⊢ 3 ∈ (0...3)
102		3ne0 12372	. . . . . . 7 ⊢ 3 ≠ 0
103		df-s3 14888	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
104	42, 103	eqtri 2765	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
105	51, 8, 53, 57, 99, 40, 101, 102, 104	vdegp1bi 29555	. . . . . 6 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = (2 + 1)
106		2p1e3 12408	. . . . . 6 ⊢ (2 + 1) = 3
107	105, 106	eqtri 2765	. . . . 5 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = 3
108		df-s4 14889	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
109	32, 108	eqtri 2765	. . . . 5 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
110	41, 8, 43, 47, 107, 30, 80, 81, 93, 94, 109	vdegp1ai 29554	. . . 4 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘0) = 3
111		df-s5 14890	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
112	22, 111	eqtri 2765	. . . 4 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
113	31, 8, 33, 37, 110, 20, 80, 81, 93, 94, 112	vdegp1ai 29554	. . 3 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘0) = 3
114		df-s6 14891	. . . 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}”⟩)
115	9, 114	eqtri 2765	. . 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}”⟩)
116	21, 8, 23, 27, 113, 4, 93, 94, 101, 102, 115	vdegp1ai 29554	. 2 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘0) = 3
117		konigsberg.v	. . 3 ⊢ 𝑉 = (0...3)
118		konigsberg.e	. . 3 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
119		konigsberg.g	. . 3 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩
120	117, 118, 119	konigsbergvtx 30265	. 2 ⊢ (Vtx‘𝐺) = (0...3)
121	117, 118, 119	konigsbergiedg 30266	. . 3 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
122	121, 12	eqtri 2765	. 2 ⊢ (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
123	5, 8, 10, 17, 116, 120, 93, 94, 101, 102, 122	vdegp1ai 29554	1 ⊢ ((VtxDeg‘𝐺)‘0) = 3

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 ⟨cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 2c2 12321 3c3 12322 ℕ₀cn0 12526 ...cfz 13547 ♯chash 14369 Word cword 14552 ++ cconcat 14608 ⟨“cs1 14633 ⟨“cs2 14880 ⟨“cs3 14881 ⟨“cs4 14882 ⟨“cs5 14883 ⟨“cs6 14884 ⟨“cs7 14885 Vtxcvtx 29013 iEdgciedg 29014 VtxDegcvtxdg 29483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-xadd 13155 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-s4 14889 df-s5 14890 df-s6 14891 df-s7 14892 df-vtx 29015 df-iedg 29016 df-vtxdg 29484

This theorem is referenced by: konigsberglem4 30274

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