konigsberglem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > konigsberglem3		Structured version Visualization version GIF version

Theorem konigsberglem3 30278

Description: Lemma 3 for konigsberg 30281: 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 7389	. . . . 5 ⊢ (0...3) ∈ V
2		s6cli 14805	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
3	2	elexi 3461	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
4	1, 3	opvtxfvi 29031	. . . 4 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
5	4	eqcomi 2743	. . 3 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6		3nn0 12417	. . . 4 ⊢ 3 ∈ ℕ₀
7		nn0fz0 13539	. . . 4 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
8	6, 7	mpbi 230	. . 3 ⊢ 3 ∈ (0...3)
9	1, 3	opiedgfvi 29032	. . . 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 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 14527	. . . 4 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
12		df-s7 14774	. . . 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}”⟩⟩
16	13, 14, 15	konigsbergssiedgw 30274	. . . 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 1463	. . 3 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
18		s5cli 14804	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
19	18	elexi 3461	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
20	1, 19	opvtxfvi 29031	. . . . . 6 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
21	20	eqcomi 2743	. . . . 5 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
22	1, 19	opiedgfvi 29032	. . . . . 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 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 14801	. . . . . 6 ⊢ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25		s5s2 14856	. . . . . 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 30274	. . . . . 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 1463	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28		s4cli 14803	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
29	28	elexi 3461	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
30	1, 29	opvtxfvi 29031	. . . . . . 7 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
31	30	eqcomi 2743	. . . . . 6 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
32	1, 29	opiedgfvi 29032	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
33	32	eqcomi 2743	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34		s3cli 14802	. . . . . . 7 ⊢ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35		s4s3 14852	. . . . . . 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 30274	. . . . . . 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 1463	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
38		s3cli 14802	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
39	38	elexi 3461	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
40	1, 39	opvtxfvi 29031	. . . . . . . 8 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
41	40	eqcomi 2743	. . . . . . 7 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
42	1, 39	opiedgfvi 29032	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
43	42	eqcomi 2743	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44		s4cli 14803	. . . . . . . 8 ⊢ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45		s3s4 14854	. . . . . . . 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 30274	. . . . . . . 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 1463	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
48		s2cli 14801	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
49	48	elexi 3461	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
50	1, 49	opvtxfvi 29031	. . . . . . . . . 10 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
51	50	eqcomi 2743	. . . . . . . . 9 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
52	1, 49	opiedgfvi 29032	. . . . . . . . . 10 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
53	52	eqcomi 2743	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54		s5cli 14804	. . . . . . . . . 10 ⊢ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55		s2s5 14855	. . . . . . . . . 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 30274	. . . . . . . . . 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 1463	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
58		s1cli 14527	. . . . . . . . . . . . 13 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
59	58	elexi 3461	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ ∈ V
60	1, 59	opvtxfvi 29031	. . . . . . . . . . 11 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
61	60	eqcomi 2743	. . . . . . . . . 10 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
62	1, 59	opiedgfvi 29032	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
63	62	eqcomi 2743	. . . . . . . . . 10 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64		s6cli 14805	. . . . . . . . . . 11 ⊢ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65		s1s6 14848	. . . . . . . . . . 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 30274	. . . . . . . . . . 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 1463	. . . . . . . . . 10 ⊢ ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
68		0ex 5250	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
69	1, 68	opvtxfvi 29031	. . . . . . . . . . . 12 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
70	69	eqcomi 2743	. . . . . . . . . . 11 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
71	1, 68	opiedgfvi 29032	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
72	71	eqcomi 2743	. . . . . . . . . . 11 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
73		wrd0 14460	. . . . . . . . . . 11 ⊢ ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
74		eqid 2734	. . . . . . . . . . . 12 ⊢ ∅ = ∅
75	70, 72	vtxdg0e 29497	. . . . . . . . . . . 12 ⊢ ((3 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0)
76	8, 74, 75	mp2an 692	. . . . . . . . . . 11 ⊢ ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0
77		0elfz 13538	. . . . . . . . . . . 12 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
78	6, 77	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ∈ (0...3)
79		3ne0 12249	. . . . . . . . . . . 12 ⊢ 3 ≠ 0
80	79	necomi 2984	. . . . . . . . . . 11 ⊢ 0 ≠ 3
81		1nn0 12415	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
82		1le3 12350	. . . . . . . . . . . 12 ⊢ 1 ≤ 3
83		elfz2nn0 13532	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
84	81, 6, 82, 83	mpbir3an 1342	. . . . . . . . . . 11 ⊢ 1 ∈ (0...3)
85		1re 11130	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
86		1lt3 12311	. . . . . . . . . . . 12 ⊢ 1 < 3
87	85, 86	ltneii 11244	. . . . . . . . . . 11 ⊢ 1 ≠ 3
88		s0s1 14843	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
89	62, 88	eqtri 2757	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
90	70, 8, 72, 73, 76, 60, 78, 80, 84, 87, 89	vdegp1ai 29559	. . . . . . . . . 10 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘3) = 0
91		2nn0 12416	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
92		2re 12217	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
93		3re 12223	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
94		2lt3 12310	. . . . . . . . . . . 12 ⊢ 2 < 3
95	92, 93, 94	ltleii 11254	. . . . . . . . . . 11 ⊢ 2 ≤ 3
96		elfz2nn0 13532	. . . . . . . . . . 11 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
97	91, 6, 95, 96	mpbir3an 1342	. . . . . . . . . 10 ⊢ 2 ∈ (0...3)
98	92, 94	ltneii 11244	. . . . . . . . . 10 ⊢ 2 ≠ 3
99		df-s2 14769	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
100	52, 99	eqtri 2757	. . . . . . . . . 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 29559	. . . . . . . . 9 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘3) = 0
102		df-s3 14770	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
103	42, 102	eqtri 2757	. . . . . . . . 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 29561	. . . . . . . 8 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = (0 + 1)
105		0p1e1 12260	. . . . . . . 8 ⊢ (0 + 1) = 1
106	104, 105	eqtri 2757	. . . . . . 7 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = 1
107		df-s4 14771	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
108	32, 107	eqtri 2757	. . . . . . 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 29559	. . . . . 6 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘3) = 1
110		df-s5 14772	. . . . . . 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 2757	. . . . . 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 29559	. . . . 5 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘3) = 1
113		df-s6 14773	. . . . . 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 2757	. . . . 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 29561	. . . 4 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = (1 + 1)
116		1p1e2 12263	. . . 4 ⊢ (1 + 1) = 2
117	115, 116	eqtri 2757	. . 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 30270	. . 3 ⊢ (Vtx‘𝐺) = (0...3)
122	118, 119, 120	konigsbergiedg 30271	. . . 4 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
123	122, 12	eqtri 2757	. . 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 29561	. 2 ⊢ ((VtxDeg‘𝐺)‘3) = (2 + 1)
125		2p1e3 12280	. 2 ⊢ (2 + 1) = 3
126	124, 125	eqtri 2757	1 ⊢ ((VtxDeg‘𝐺)‘3) = 3

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2113 {crab 3397 Vcvv 3438 ∖ cdif 3896 ∅c0 4283 𝒫 cpw 4552 {csn 4578 {cpr 4580 ⟨cop 4584 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 0cc0 11024 1c1 11025 + caddc 11027 ≤ cle 11165 2c2 12198 3c3 12199 ℕ₀cn0 12399 ...cfz 13421 ♯chash 14251 Word cword 14434 ++ cconcat 14491 ⟨“cs1 14517 ⟨“cs2 14762 ⟨“cs3 14763 ⟨“cs4 14764 ⟨“cs5 14765 ⟨“cs6 14766 ⟨“cs7 14767 Vtxcvtx 29018 iEdgciedg 29019 VtxDegcvtxdg 29488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-dju 9811 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-xadd 13025 df-fz 13422 df-fzo 13569 df-hash 14252 df-word 14435 df-concat 14492 df-s1 14518 df-s2 14769 df-s3 14770 df-s4 14771 df-s5 14772 df-s6 14773 df-s7 14774 df-vtx 29020 df-iedg 29021 df-vtxdg 29489

This theorem is referenced by: konigsberglem4 30279

W3C validator