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Theorem konigsberglem2 28518

Description: Lemma 2 for konigsberg 28522: 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.

Step	Hyp	Ref	Expression
1		ovex 7288	. . . 4 ⊢ (0...3) ∈ V
2		s6cli 14525	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
3	2	elexi 3441	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
4	1, 3	opvtxfvi 27282	. . 3 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
5	4	eqcomi 2747	. 2 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6		1nn0 12179	. . 3 ⊢ 1 ∈ ℕ₀
7		3nn0 12181	. . 3 ⊢ 3 ∈ ℕ₀
8		1le3 12115	. . 3 ⊢ 1 ≤ 3
9		elfz2nn0 13276	. . 3 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
10	6, 7, 8, 9	mpbir3an 1339	. 2 ⊢ 1 ∈ (0...3)
11	1, 3	opiedgfvi 27283	. . 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}”⟩
12	11	eqcomi 2747	. 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 14238	. . 3 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
14		df-s7 14494	. . 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 2738	. . . 4 ⊢ (0...3) = (0...3)
16		eqid 2738	. . . 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 2738	. . . 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}”⟩⟩
18	15, 16, 17	konigsbergssiedgw 28515	. . 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})
19	2, 13, 14, 18	mp3an 1459	. 2 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
20		s5cli 14524	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
21	20	elexi 3441	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
22	1, 21	opvtxfvi 27282	. . . 4 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
23	22	eqcomi 2747	. . 3 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
24	1, 21	opiedgfvi 27283	. . . 4 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
25	24	eqcomi 2747	. . 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 14521	. . . 4 ⊢ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
27		s5s2 14576	. . . 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}”⟩)
28	15, 16, 17	konigsbergssiedgw 28515	. . . 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})
29	20, 26, 27, 28	mp3an 1459	. . 3 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
30		s4cli 14523	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
31	30	elexi 3441	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
32	1, 31	opvtxfvi 27282	. . . . . 6 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
33	32	eqcomi 2747	. . . . 5 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34	1, 31	opiedgfvi 27283	. . . . . 6 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
35	34	eqcomi 2747	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
36		s3cli 14522	. . . . . 6 ⊢ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
37		s4s3 14572	. . . . . 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}”⟩)
38	15, 16, 17	konigsbergssiedgw 28515	. . . . . 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})
39	30, 36, 37, 38	mp3an 1459	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
40		s3cli 14522	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
41	40	elexi 3441	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
42	1, 41	opvtxfvi 27282	. . . . . . . 8 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
43	42	eqcomi 2747	. . . . . . 7 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44	1, 41	opiedgfvi 27283	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
45	44	eqcomi 2747	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
46		s4cli 14523	. . . . . . . 8 ⊢ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
47		s3s4 14574	. . . . . . . 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}”⟩)
48	15, 16, 17	konigsbergssiedgw 28515	. . . . . . . 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})
49	40, 46, 47, 48	mp3an 1459	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
50		s2cli 14521	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
51	50	elexi 3441	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
52	1, 51	opvtxfvi 27282	. . . . . . . . 9 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
53	52	eqcomi 2747	. . . . . . . 8 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54	1, 51	opiedgfvi 27283	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
55	54	eqcomi 2747	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
56		s5cli 14524	. . . . . . . . 9 ⊢ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
57		s2s5 14575	. . . . . . . . 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}”⟩)
58	15, 16, 17	konigsbergssiedgw 28515	. . . . . . . . 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})
59	50, 56, 57, 58	mp3an 1459	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
60		s1cli 14238	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
61	60	elexi 3441	. . . . . . . . . . 11 ⊢ ⟨“{0, 1}”⟩ ∈ V
62	1, 61	opvtxfvi 27282	. . . . . . . . . 10 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
63	62	eqcomi 2747	. . . . . . . . 9 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64	1, 61	opiedgfvi 27283	. . . . . . . . . 10 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
65	64	eqcomi 2747	. . . . . . . . 9 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
66		s6cli 14525	. . . . . . . . . 10 ⊢ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
67		s1s6 14568	. . . . . . . . . 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}”⟩)
68	15, 16, 17	konigsbergssiedgw 28515	. . . . . . . . . 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})
69	60, 66, 67, 68	mp3an 1459	. . . . . . . . 9 ⊢ ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
70		0ex 5226	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
71	1, 70	opvtxfvi 27282	. . . . . . . . . . . 12 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
72	71	eqcomi 2747	. . . . . . . . . . 11 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
73	1, 70	opiedgfvi 27283	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
74	73	eqcomi 2747	. . . . . . . . . . 11 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
75		wrd0 14170	. . . . . . . . . . 11 ⊢ ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
76		eqid 2738	. . . . . . . . . . . 12 ⊢ ∅ = ∅
77	72, 74	vtxdg0e 27744	. . . . . . . . . . . 12 ⊢ ((1 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0)
78	10, 76, 77	mp2an 688	. . . . . . . . . . 11 ⊢ ((VtxDeg‘⟨(0...3), ∅⟩)‘1) = 0
79		0elfz 13282	. . . . . . . . . . . 12 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
80	7, 79	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...3)
81		0ne1 11974	. . . . . . . . . . 11 ⊢ 0 ≠ 1
82		s0s1 14563	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
83	64, 82	eqtri 2766	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
84	72, 10, 74, 75, 78, 62, 80, 81, 83	vdegp1ci 27808	. . . . . . . . . 10 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = (0 + 1)
85		0p1e1 12025	. . . . . . . . . 10 ⊢ (0 + 1) = 1
86	84, 85	eqtri 2766	. . . . . . . . 9 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘1) = 1
87		2nn0 12180	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
88		2re 11977	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
89		3re 11983	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
90		2lt3 12075	. . . . . . . . . . 11 ⊢ 2 < 3
91	88, 89, 90	ltleii 11028	. . . . . . . . . 10 ⊢ 2 ≤ 3
92		elfz2nn0 13276	. . . . . . . . . 10 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
93	87, 7, 91, 92	mpbir3an 1339	. . . . . . . . 9 ⊢ 2 ∈ (0...3)
94		1ne2 12111	. . . . . . . . . 10 ⊢ 1 ≠ 2
95	94	necomi 2997	. . . . . . . . 9 ⊢ 2 ≠ 1
96		df-s2 14489	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
97	54, 96	eqtri 2766	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
98	63, 10, 65, 69, 86, 52, 80, 81, 93, 95, 97	vdegp1ai 27806	. . . . . . . 8 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘1) = 1
99		nn0fz0 13283	. . . . . . . . 9 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
100	7, 99	mpbi 229	. . . . . . . 8 ⊢ 3 ∈ (0...3)
101		1re 10906	. . . . . . . . 9 ⊢ 1 ∈ ℝ
102		1lt3 12076	. . . . . . . . 9 ⊢ 1 < 3
103	101, 102	gtneii 11017	. . . . . . . 8 ⊢ 3 ≠ 1
104		df-s3 14490	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
105	44, 104	eqtri 2766	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
106	53, 10, 55, 59, 98, 42, 80, 81, 100, 103, 105	vdegp1ai 27806	. . . . . . 7 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘1) = 1
107		df-s4 14491	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
108	34, 107	eqtri 2766	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
109	43, 10, 45, 49, 106, 32, 93, 95, 108	vdegp1bi 27807	. . . . . 6 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = (1 + 1)
110		1p1e2 12028	. . . . . 6 ⊢ (1 + 1) = 2
111	109, 110	eqtri 2766	. . . . 5 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘1) = 2
112		df-s5 14492	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
113	24, 112	eqtri 2766	. . . . 5 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
114	33, 10, 35, 39, 111, 22, 93, 95, 113	vdegp1bi 27807	. . . 4 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = (2 + 1)
115		2p1e3 12045	. . . 4 ⊢ (2 + 1) = 3
116	114, 115	eqtri 2766	. . 3 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘1) = 3
117		df-s6 14493	. . . 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}”⟩)
118	11, 117	eqtri 2766	. . 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}”⟩)
119	23, 10, 25, 29, 116, 4, 93, 95, 100, 103, 118	vdegp1ai 27806	. 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⟩
123	120, 121, 122	konigsbergvtx 28511	. 2 ⊢ (Vtx‘𝐺) = (0...3)
124	120, 121, 122	konigsbergiedg 28512	. . 3 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
125	124, 14	eqtri 2766	. 2 ⊢ (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
126	5, 10, 12, 19, 119, 123, 93, 95, 100, 103, 125	vdegp1ai 27806	1 ⊢ ((VtxDeg‘𝐺)‘1) = 3

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 ∖ cdif 3880 ∅c0 4253 𝒫 cpw 4530 {csn 4558 {cpr 4560 ⟨cop 4564 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 0cc0 10802 1c1 10803 + caddc 10805 ≤ cle 10941 2c2 11958 3c3 11959 ℕ₀cn0 12163 ...cfz 13168 ♯chash 13972 Word cword 14145 ++ cconcat 14201 ⟨“cs1 14228 ⟨“cs2 14482 ⟨“cs3 14483 ⟨“cs4 14484 ⟨“cs5 14485 ⟨“cs6 14486 ⟨“cs7 14487 Vtxcvtx 27269 iEdgciedg 27270 VtxDegcvtxdg 27735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-xadd 12778 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-s2 14489 df-s3 14490 df-s4 14491 df-s5 14492 df-s6 14493 df-s7 14494 df-vtx 27271 df-iedg 27272 df-vtxdg 27736

This theorem is referenced by: konigsberglem4 28520

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