Theorem konigsberglem3 30082

Description: Lemma 3 for konigsberg 30085: 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.

Step	Hyp	Ref	Expression
1		ovex 7457	. . . . 5 ⊢ (0...3) ∈ V
2		s6cli 14873	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
3	2	elexi 3491	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
4	1, 3	opvtxfvi 28840	. . . 4 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
5	4	eqcomi 2736	. . 3 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6		3nn0 12526	. . . 4 ⊢ 3 ∈ ℕ0
7		nn0fz0 13637	. . . 4 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
8	6, 7	mpbi 229	. . 3 ⊢ 3 ∈ (0...3)
9	1, 3	opiedgfvi 28841	. . . 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}”⟩
10	9	eqcomi 2736	. . 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 14593	. . . 4 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
12		df-s7 14842	. . . 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 2727	. . . . 5 ⊢ (0...3) = (0...3)
14		eqid 2727	. . . . 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 2727	. . . . 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}”⟩⟩
16	13, 14, 15	konigsbergssiedgw 30078	. . . 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})
17	2, 11, 12, 16	mp3an 1457	. . 3 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
18		s5cli 14872	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
19	18	elexi 3491	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
20	1, 19	opvtxfvi 28840	. . . . . 6 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
21	20	eqcomi 2736	. . . . 5 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
22	1, 19	opiedgfvi 28841	. . . . . 6 ⊢ (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 2736	. . . . 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 14869	. . . . . 6 ⊢ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25		s5s2 14924	. . . . . 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}”⟩)
26	13, 14, 15	konigsbergssiedgw 30078	. . . . . 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})
27	18, 24, 25, 26	mp3an 1457	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28		s4cli 14871	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
29	28	elexi 3491	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
30	1, 29	opvtxfvi 28840	. . . . . . 7 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
31	30	eqcomi 2736	. . . . . 6 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
32	1, 29	opiedgfvi 28841	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
33	32	eqcomi 2736	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34		s3cli 14870	. . . . . . 7 ⊢ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35		s4s3 14920	. . . . . . 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}”⟩)
36	13, 14, 15	konigsbergssiedgw 30078	. . . . . . 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})
37	28, 34, 35, 36	mp3an 1457	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
38		s3cli 14870	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
39	38	elexi 3491	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
40	1, 39	opvtxfvi 28840	. . . . . . . 8 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
41	40	eqcomi 2736	. . . . . . 7 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
42	1, 39	opiedgfvi 28841	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
43	42	eqcomi 2736	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44		s4cli 14871	. . . . . . . 8 ⊢ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45		s3s4 14922	. . . . . . . 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}”⟩)
46	13, 14, 15	konigsbergssiedgw 30078	. . . . . . . 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})
47	38, 44, 45, 46	mp3an 1457	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
48		s2cli 14869	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
49	48	elexi 3491	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
50	1, 49	opvtxfvi 28840	. . . . . . . . . 10 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
51	50	eqcomi 2736	. . . . . . . . 9 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
52	1, 49	opiedgfvi 28841	. . . . . . . . . 10 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
53	52	eqcomi 2736	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54		s5cli 14872	. . . . . . . . . 10 ⊢ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55		s2s5 14923	. . . . . . . . . 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}”⟩)
56	13, 14, 15	konigsbergssiedgw 30078	. . . . . . . . . 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})
57	48, 54, 55, 56	mp3an 1457	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
58		s1cli 14593	. . . . . . . . . . . . 13 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
59	58	elexi 3491	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ ∈ V
60	1, 59	opvtxfvi 28840	. . . . . . . . . . 11 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
61	60	eqcomi 2736	. . . . . . . . . 10 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
62	1, 59	opiedgfvi 28841	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
63	62	eqcomi 2736	. . . . . . . . . 10 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64		s6cli 14873	. . . . . . . . . . 11 ⊢ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65		s1s6 14916	. . . . . . . . . . 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}”⟩)
66	13, 14, 15	konigsbergssiedgw 30078	. . . . . . . . . . 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})
67	58, 64, 65, 66	mp3an 1457	. . . . . . . . . 10 ⊢ ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
68		0ex 5309	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
69	1, 68	opvtxfvi 28840	. . . . . . . . . . . 12 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
70	69	eqcomi 2736	. . . . . . . . . . 11 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
71	1, 68	opiedgfvi 28841	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
72	71	eqcomi 2736	. . . . . . . . . . 11 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
73		wrd0 14527	. . . . . . . . . . 11 ⊢ ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
74		eqid 2727	. . . . . . . . . . . 12 ⊢ ∅ = ∅
75	70, 72	vtxdg0e 29306	. . . . . . . . . . . 12 ⊢ ((3 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0)
76	8, 74, 75	mp2an 690	. . . . . . . . . . 11 ⊢ ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0
77		0elfz 13636	. . . . . . . . . . . 12 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3))
78	6, 77	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...3)
79		3ne0 12354	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
80	79	necomi 2991	. . . . . . . . . . 11 ⊢ 0 ≠ 3
81		1nn0 12524	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ0
82		1le3 12460	. . . . . . . . . . . 12 ⊢ 1 ≤ 3
83		elfz2nn0 13630	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
84	81, 6, 82, 83	mpbir3an 1338	. . . . . . . . . . 11 ⊢ 1 ∈ (0...3)
85		1re 11250	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
86		1lt3 12421	. . . . . . . . . . . 12 ⊢ 1 < 3
87	85, 86	ltneii 11363	. . . . . . . . . . 11 ⊢ 1 ≠ 3
88		s0s1 14911	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
89	62, 88	eqtri 2755	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
90	70, 8, 72, 73, 76, 60, 78, 80, 84, 87, 89	vdegp1ai 29368	. . . . . . . . . 10 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘3) = 0
91		2nn0 12525	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ0
92		2re 12322	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
93		3re 12328	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
94		2lt3 12420	. . . . . . . . . . . 12 ⊢ 2 < 3
95	92, 93, 94	ltleii 11373	. . . . . . . . . . 11 ⊢ 2 ≤ 3
96		elfz2nn0 13630	. . . . . . . . . . 11 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
97	91, 6, 95, 96	mpbir3an 1338	. . . . . . . . . 10 ⊢ 2 ∈ (0...3)
98	92, 94	ltneii 11363	. . . . . . . . . 10 ⊢ 2 ≠ 3
99		df-s2 14837	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
100	52, 99	eqtri 2755	. . . . . . . . . 10 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
101	61, 8, 63, 67, 90, 50, 78, 80, 97, 98, 100	vdegp1ai 29368	. . . . . . . . 9 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘3) = 0
102		df-s3 14838	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
103	42, 102	eqtri 2755	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
104	51, 8, 53, 57, 101, 40, 78, 80, 103	vdegp1ci 29370	. . . . . . . 8 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = (0 + 1)
105		0p1e1 12370	. . . . . . . 8 ⊢ (0 + 1) = 1
106	104, 105	eqtri 2755	. . . . . . 7 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = 1
107		df-s4 14839	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
108	32, 107	eqtri 2755	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
109	41, 8, 43, 47, 106, 30, 84, 87, 97, 98, 108	vdegp1ai 29368	. . . . . 6 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘3) = 1
110		df-s5 14840	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
111	22, 110	eqtri 2755	. . . . . 6 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
112	31, 8, 33, 37, 109, 20, 84, 87, 97, 98, 111	vdegp1ai 29368	. . . . 5 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘3) = 1
113		df-s6 14841	. . . . . 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}”⟩)
114	9, 113	eqtri 2755	. . . . 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}”⟩)
115	21, 8, 23, 27, 112, 4, 97, 98, 114	vdegp1ci 29370	. . . 4 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = (1 + 1)
116		1p1e2 12373	. . . 4 ⊢ (1 + 1) = 2
117	115, 116	eqtri 2755	. . 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⟩
121	118, 119, 120	konigsbergvtx 30074	. . 3 ⊢ (Vtx‘𝐺) = (0...3)
122	118, 119, 120	konigsbergiedg 30075	. . . 4 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
123	122, 12	eqtri 2755	. . 3 ⊢ (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
124	5, 8, 10, 17, 117, 121, 97, 98, 123	vdegp1ci 29370	. 2 ⊢ ((VtxDeg‘𝐺)‘3) = (2 + 1)
125		2p1e3 12390	. 2 ⊢ (2 + 1) = 3
126	124, 125	eqtri 2755	1 ⊢ ((VtxDeg‘𝐺)‘3) = 3

Syntax hints:   = wceq 1533   ∈ wcel 2098  {crab 3428  Vcvv 3471   ∖ cdif 3944  ∅c0 4324  𝒫 cpw 4604  {csn 4630  {cpr 4632  ⟨cop 4636   class class class wbr 5150  ‘cfv 6551  (class class class)co 7424  0cc0 11144  1c1 11145   + caddc 11147   ≤ cle 11285  2c2 12303  3c3 12304  ℕ₀cn0 12508  ...cfz 13522  ♯chash 14327  Word cword 14502   ++ cconcat 14558  ⟨“cs1 14583  ⟨“cs2 14830  ⟨“cs3 14831  ⟨“cs4 14832  ⟨“cs5 14833  ⟨“cs6 14834  ⟨“cs7 14835  Vtxcvtx 28827  iEdgciedg 28828  VtxDegcvtxdg 29297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-oadd 8495  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-dju 9930  df-card 9968  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-n0 12509  df-xnn0 12581  df-z 12595  df-uz 12859  df-xadd 13131  df-fz 13523  df-fzo 13666  df-hash 14328  df-word 14503  df-concat 14559  df-s1 14584  df-s2 14837  df-s3 14838  df-s4 14839  df-s5 14840  df-s6 14841  df-s7 14842  df-vtx 28829  df-iedg 28830  df-vtxdg 29298

This theorem is referenced by:  konigsberglem4  30083