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Theorem konigsberglem1 30718

Description: Lemma 1 for konigsberg 30723: 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 7449	. . . 4 ⊢ (0...3) ∈ V
2		s6cli 14957	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
3	2	elexi 3475	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
4	1, 3	opvtxfvi 29452	. . 3 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
5	4	eqcomi 2771	. 2 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6		3nn0 12549	. . 3 ⊢ 3 ∈ ℕ₀
7		0elfz 13681	. . 3 ⊢ (3 ∈ ℕ₀ → 0 ∈ (0...3))
8	6, 7	ax-mp 5	. 2 ⊢ 0 ∈ (0...3)
9	1, 3	opiedgfvi 29453	. . 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 2771	. 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 14674	. . 3 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
12		df-s7 14926	. . 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 2762	. . . 4 ⊢ (0...3) = (0...3)
14		eqid 2762	. . . 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 2762	. . . 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 30716	. . 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 1490	. 2 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
18		s5cli 14956	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
19	18	elexi 3475	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
20	1, 19	opvtxfvi 29452	. . . 4 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
21	20	eqcomi 2771	. . 3 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
22	1, 19	opiedgfvi 29453	. . . 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 2771	. . 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 14953	. . . 4 ⊢ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25		s5s2 15008	. . . 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 30716	. . . 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 1490	. . 3 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
28		s4cli 14955	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
29	28	elexi 3475	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
30	1, 29	opvtxfvi 29452	. . . . 5 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
31	30	eqcomi 2771	. . . 4 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
32	1, 29	opiedgfvi 29453	. . . . 5 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
33	32	eqcomi 2771	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34		s3cli 14954	. . . . 5 ⊢ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35		s4s3 15004	. . . . 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 30716	. . . . 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 1490	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
38		s3cli 14954	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
39	38	elexi 3475	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
40	1, 39	opvtxfvi 29452	. . . . . 6 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
41	40	eqcomi 2771	. . . . 5 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
42	1, 39	opiedgfvi 29453	. . . . . 6 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
43	42	eqcomi 2771	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44		s4cli 14955	. . . . . 6 ⊢ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45		s3s4 15006	. . . . . 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 30716	. . . . . 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 1490	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
48		s2cli 14953	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
49	48	elexi 3475	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
50	1, 49	opvtxfvi 29452	. . . . . . . 8 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
51	50	eqcomi 2771	. . . . . . 7 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
52	1, 49	opiedgfvi 29453	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
53	52	eqcomi 2771	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54		s5cli 14956	. . . . . . . 8 ⊢ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55		s2s5 15007	. . . . . . . 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 30716	. . . . . . . 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 1490	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
58		s1cli 14674	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
59	58	elexi 3475	. . . . . . . . . . 11 ⊢ ⟨“{0, 1}”⟩ ∈ V
60	1, 59	opvtxfvi 29452	. . . . . . . . . 10 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
61	60	eqcomi 2771	. . . . . . . . 9 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
62	1, 59	opiedgfvi 29453	. . . . . . . . . 10 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
63	62	eqcomi 2771	. . . . . . . . 9 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64		s6cli 14957	. . . . . . . . . 10 ⊢ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65		s1s6 15000	. . . . . . . . . 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 30716	. . . . . . . . . 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 1490	. . . . . . . . 9 ⊢ ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
68		0ex 5268	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
69	1, 68	opvtxfvi 29452	. . . . . . . . . . . 12 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
70	69	eqcomi 2771	. . . . . . . . . . 11 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
71	1, 68	opiedgfvi 29453	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
72	71	eqcomi 2771	. . . . . . . . . . 11 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
73		wrd0 14606	. . . . . . . . . . 11 ⊢ ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}
74		eqid 2762	. . . . . . . . . . . 12 ⊢ ∅ = ∅
75	70, 72	vtxdg0e 29920	. . . . . . . . . . . 12 ⊢ ((0 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0)
76	8, 74, 75	mp2an 705	. . . . . . . . . . 11 ⊢ ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0
77		1nn0 12547	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
78		1le3 12482	. . . . . . . . . . . 12 ⊢ 1 ≤ 3
79		elfz2nn0 13675	. . . . . . . . . . . 12 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 1 ≤ 3))
80	77, 6, 78, 79	mpbir3an 1360	. . . . . . . . . . 11 ⊢ 1 ∈ (0...3)
81		ax-1ne0 11196	. . . . . . . . . . 11 ⊢ 1 ≠ 0
82		s0s1 14995	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
83	62, 82	eqtri 2785	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
84	70, 8, 72, 73, 76, 60, 80, 81, 83	vdegp1bi 29983	. . . . . . . . . 10 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = (0 + 1)
85		0p1e1 12388	. . . . . . . . . 10 ⊢ (0 + 1) = 1
86	84, 85	eqtri 2785	. . . . . . . . 9 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = 1
87		2nn0 12548	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
88		2re 12342	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
89		3re 12348	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
90		2lt3 12441	. . . . . . . . . . 11 ⊢ 2 < 3
91	88, 89, 90	ltleii 11360	. . . . . . . . . 10 ⊢ 2 ≤ 3
92		elfz2nn0 13675	. . . . . . . . . 10 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀ ∧ 2 ≤ 3))
93	87, 6, 91, 92	mpbir3an 1360	. . . . . . . . 9 ⊢ 2 ∈ (0...3)
94		2ne0 12374	. . . . . . . . 9 ⊢ 2 ≠ 0
95		df-s2 14921	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
96	52, 95	eqtri 2785	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
97	61, 8, 63, 67, 86, 50, 93, 94, 96	vdegp1bi 29983	. . . . . . . 8 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = (1 + 1)
98		1p1e2 12391	. . . . . . . 8 ⊢ (1 + 1) = 2
99	97, 98	eqtri 2785	. . . . . . 7 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = 2
100		nn0fz0 13682	. . . . . . . 8 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (0...3))
101	6, 100	mpbi 233	. . . . . . 7 ⊢ 3 ∈ (0...3)
102		3ne0 12377	. . . . . . 7 ⊢ 3 ≠ 0
103		df-s3 14922	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
104	42, 103	eqtri 2785	. . . . . . 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 29983	. . . . . 6 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = (2 + 1)
106		2p1e3 12409	. . . . . 6 ⊢ (2 + 1) = 3
107	105, 106	eqtri 2785	. . . . 5 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = 3
108		df-s4 14923	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
109	32, 108	eqtri 2785	. . . . 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 29982	. . . 4 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘0) = 3
111		df-s5 14924	. . . . 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 2785	. . . 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 29982	. . 3 ⊢ ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘0) = 3
114		df-s6 14925	. . . 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 2785	. . 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 29982	. 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 30712	. 2 ⊢ (Vtx‘𝐺) = (0...3)
121	117, 118, 119	konigsbergiedg 30713	. . 3 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
122	121, 12	eqtri 2785	. 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 29982	1 ⊢ ((VtxDeg‘𝐺)‘0) = 3

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 {crab 3414 Vcvv 3453 ∖ cdif 3899 ∅c0 4282 𝒫 cpw 4560 {csn 4587 {cpr 4589 ⟨cop 4593 class class class wbr 5107 ‘cfv 6537 (class class class)co 7416 0cc0 11127 1c1 11128 + caddc 11130 ≤ cle 11271 2c2 12322 3c3 12323 ℕ₀cn0 12531 ...cfz 13563 ♯chash 14396 Word cword 14580 ++ cconcat 14637 ⟨“cs1 14664 ⟨“cs2 14914 ⟨“cs3 14915 ⟨“cs4 14916 ⟨“cs5 14917 ⟨“cs6 14918 ⟨“cs7 14919 Vtxcvtx 29439 iEdgciedg 29440 VtxDegcvtxdg 29911

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-n0 12532 df-xnn0 12605 df-z 12619 df-uz 12891 df-xadd 13166 df-fz 13564 df-fzo 13712 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-s2 14921 df-s3 14922 df-s4 14923 df-s5 14924 df-s6 14925 df-s7 14926 df-vtx 29441 df-iedg 29442 df-vtxdg 29912

This theorem is used by: konigsberglem4 30721

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