Theorem konigsberglem1 16643

Description: Lemma 1 for konigsberg 16648: 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		0z 9634	. . . . . . 7 ⊢ 0 ∈ ℤ
2		3z 9652	. . . . . . 7 ⊢ 3 ∈ ℤ
3		fzfig 10845	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0...3) ∈ Fin)
4	1, 2, 3	mp2an 430	. . . . . 6 ⊢ (0...3) ∈ Fin
5	4	elexi 2834	. . . . 5 ⊢ (0...3) ∈ V
6		1zzd 9650	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℤ)
7		prexg 4344	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ) → {0, 1} ∈ V)
8	1, 6, 7	sylancr 418	. . . . . . . 8 ⊢ (⊤ → {0, 1} ∈ V)
9	1	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 0 ∈ ℤ)
10		2z 9651	. . . . . . . . 9 ⊢ 2 ∈ ℤ
11		prexg 4344	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 2 ∈ ℤ) → {0, 2} ∈ V)
12	9, 10, 11	sylancl 417	. . . . . . . 8 ⊢ (⊤ → {0, 2} ∈ V)
13		prexg 4344	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → {0, 3} ∈ V)
14	9, 2, 13	sylancl 417	. . . . . . . 8 ⊢ (⊤ → {0, 3} ∈ V)
15		prexg 4344	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 2 ∈ ℤ) → {1, 2} ∈ V)
16	6, 10, 15	sylancl 417	. . . . . . . 8 ⊢ (⊤ → {1, 2} ∈ V)
17	10	a1i 9	. . . . . . . . 9 ⊢ (⊤ → 2 ∈ ℤ)
18		prexg 4344	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 3 ∈ ℤ) → {2, 3} ∈ V)
19	17, 2, 18	sylancl 417	. . . . . . . 8 ⊢ (⊤ → {2, 3} ∈ V)
20	8, 12, 14, 16, 16, 19	s6cld 11532	. . . . . . 7 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V)
21	20	mptru 1411	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
22	21	elexi 2834	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
23	5, 22	opvtxfvi 16182	. . . 4 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
24	23	eqcomi 2242	. . 3 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
25		3nn0 9560	. . . . 5 ⊢ 3 ∈ ℕ0
26		0elfz 10503	. . . . 5 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3))
27	25, 26	ax-mp 5	. . . 4 ⊢ 0 ∈ (0...3)
28	27	a1i 9	. . 3 ⊢ (⊤ → 0 ∈ (0...3))
29	5, 22	opiedgfvi 16183	. . . 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}”⟩
30	29	eqcomi 2242	. . 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}”⟩⟩)
31	19	s1cld 11368	. . . . . 6 ⊢ (⊤ → ⟨“{2, 3}”⟩ ∈ Word V)
32	31	mptru 1411	. . . . 5 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
33		df-s7 11511	. . . . 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}”⟩)
34		eqid 2238	. . . . . 6 ⊢ (0...3) = (0...3)
35		eqid 2238	. . . . . 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}”⟩
36		eqid 2238	. . . . . 6 ⊢ ⟨(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}”⟩⟩
37	34, 35, 36	konigsbergssiedgwen 16641	. . . . 5 ⊢ ((⟨“{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 ≈ 1o ∨ 𝑥 ≈ 2o)})
38	21, 32, 33, 37	mp3an 1378	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
39	38	a1i 9	. . 3 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
40	8, 12, 14, 16, 16	s5cld 11531	. . . . . . . 8 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V)
41	40	mptru 1411	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
42	41	elexi 2834	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
43	5, 42	opvtxfvi 16182	. . . . 5 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
44	43	eqcomi 2242	. . . 4 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
45	5, 42	opiedgfvi 16183	. . . . 5 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
46	45	eqcomi 2242	. . . 4 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
47	19, 19	s2cld 11528	. . . . 5 ⊢ (⊤ → ⟨“{2, 3} {2, 3}”⟩ ∈ Word V)
48	8, 12, 14, 16, 16, 19, 19	s5s2d 11555	. . . . 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}”⟩))
49	34, 35, 36	konigsbergssiedgwen 16641	. . . . 5 ⊢ ((⟨“{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 ≈ 1o ∨ 𝑥 ≈ 2o)})
50	40, 47, 48, 49	syl3anc 1278	. . . 4 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
51	8, 12, 14, 16	s4cld 11530	. . . . . . . . 9 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V)
52	51	mptru 1411	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
53	52	elexi 2834	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
54	5, 53	opvtxfvi 16182	. . . . . 6 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
55	54	eqcomi 2242	. . . . 5 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
56	5, 53	opiedgfvi 16183	. . . . . 6 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
57	56	eqcomi 2242	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
58	16, 19, 19	s3cld 11529	. . . . . 6 ⊢ (⊤ → ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
59	8, 12, 14, 16, 16, 19, 19	s4s3d 11552	. . . . . 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}”⟩))
60	34, 35, 36	konigsbergssiedgwen 16641	. . . . . 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 ≈ 1o ∨ 𝑥 ≈ 2o)})
61	51, 58, 59, 60	syl3anc 1278	. . . . 5 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
62	8	mptru 1411	. . . . . . . . . 10 ⊢ {0, 1} ∈ V
63	1, 10, 11	mp2an 430	. . . . . . . . . 10 ⊢ {0, 2} ∈ V
64	1, 2, 13	mp2an 430	. . . . . . . . . 10 ⊢ {0, 3} ∈ V
65		s3cl 11536	. . . . . . . . . 10 ⊢ (({0, 1} ∈ V ∧ {0, 2} ∈ V ∧ {0, 3} ∈ V) → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V)
66	62, 63, 64, 65	mp3an 1378	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
67	66	elexi 2834	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
68	5, 67	opvtxfvi 16182	. . . . . . 7 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
69	68	eqcomi 2242	. . . . . 6 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
70	5, 67	opiedgfvi 16183	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
71	70	eqcomi 2242	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
72	16, 16, 19, 19	s4cld 11530	. . . . . . . . 9 ⊢ (⊤ → ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
73	72	mptru 1411	. . . . . . . 8 ⊢ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
74	8, 12, 14, 16, 16, 19, 19	s3s4d 11553	. . . . . . . . 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}”⟩))
75	74	mptru 1411	. . . . . . . 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}”⟩)
76	34, 35, 36	konigsbergssiedgwen 16641	. . . . . . . 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 ≈ 1o ∨ 𝑥 ≈ 2o)})
77	66, 73, 75, 76	mp3an 1378	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
78	77	a1i 9	. . . . . 6 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
79		s2cl 11535	. . . . . . . . . . . 12 ⊢ (({0, 1} ∈ V ∧ {0, 2} ∈ V) → ⟨“{0, 1} {0, 2}”⟩ ∈ Word V)
80	62, 63, 79	mp2an 430	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
81	80	elexi 2834	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
82	5, 81	opvtxfvi 16182	. . . . . . . . 9 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
83	82	eqcomi 2242	. . . . . . . 8 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
84	5, 81	opiedgfvi 16183	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
85	84	eqcomi 2242	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
86		s1fv 11372	. . . . . . . . . . . 12 ⊢ ({0, 1} ∈ V → (⟨“{0, 1}”⟩‘0) = {0, 1})
87	62, 86	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨“{0, 1}”⟩‘0) = {0, 1}
88		1nn 9294	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
89		s1cl 11367	. . . . . . . . . . . . . . 15 ⊢ ({0, 1} ∈ V → ⟨“{0, 1}”⟩ ∈ Word V)
90	62, 89	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
91	12, 14, 16, 16, 19, 19	s6cld 11532	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
92	91	mptru 1411	. . . . . . . . . . . . . 14 ⊢ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
93	8, 12, 14, 16, 16, 19, 19	s1s6d 11548	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ⟨“{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}”⟩))
94	93	mptru 1411	. . . . . . . . . . . . . 14 ⊢ ⟨“{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}”⟩)
95	34, 35, 36	konigsbergssiedgwen 16641	. . . . . . . . . . . . . 14 ⊢ ((⟨“{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 ≈ 1o ∨ 𝑥 ≈ 2o)})
96	90, 92, 94, 95	mp3an 1378	. . . . . . . . . . . . 13 ⊢ ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
97		s1leng 11370	. . . . . . . . . . . . . 14 ⊢ ({0, 1} ∈ V → (♯‘⟨“{0, 1}”⟩) = 1)
98	62, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (♯‘⟨“{0, 1}”⟩) = 1
99	96, 98	pm3.2i 272	. . . . . . . . . . . 12 ⊢ (⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ∧ (♯‘⟨“{0, 1}”⟩) = 1)
100		fstwrdne0 11322	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℕ ∧ (⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ∧ (♯‘⟨“{0, 1}”⟩) = 1)) → (⟨“{0, 1}”⟩‘0) ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
101	88, 99, 100	mp2an 430	. . . . . . . . . . 11 ⊢ (⟨“{0, 1}”⟩‘0) ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
102	87, 101	eqeltrri 2312	. . . . . . . . . 10 ⊢ {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
103	102	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {0, 1} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
104		2nn0 9559	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ0
105		2re 9353	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
106		3re 9357	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℝ
107		2lt3 9454	. . . . . . . . . . . . . 14 ⊢ 2 < 3
108	105, 106, 107	ltleii 8418	. . . . . . . . . . . . 13 ⊢ 2 ≤ 3
109		elfz2nn0 10497	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
110	104, 25, 108, 109	mpbir3an 1210	. . . . . . . . . . . 12 ⊢ 2 ∈ (0...3)
111		prelpwi 4349	. . . . . . . . . . . 12 ⊢ ((0 ∈ (0...3) ∧ 2 ∈ (0...3)) → {0, 2} ∈ 𝒫 (0...3))
112	27, 110, 111	mp2an 430	. . . . . . . . . . 11 ⊢ {0, 2} ∈ 𝒫 (0...3)
113		0ne2 9489	. . . . . . . . . . . . 13 ⊢ 0 ≠ 2
114		pr2ne 7528	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 2 ∈ ℤ) → ({0, 2} ≈ 2o ↔ 0 ≠ 2))
115	1, 10, 114	mp2an 430	. . . . . . . . . . . . 13 ⊢ ({0, 2} ≈ 2o ↔ 0 ≠ 2)
116	113, 115	mpbir 146	. . . . . . . . . . . 12 ⊢ {0, 2} ≈ 2o
117	116	olci 744	. . . . . . . . . . 11 ⊢ ({0, 2} ≈ 1o ∨ {0, 2} ≈ 2o)
118		breq1 4128	. . . . . . . . . . . . 13 ⊢ (𝑥 = {0, 2} → (𝑥 ≈ 1o ↔ {0, 2} ≈ 1o))
119		breq1 4128	. . . . . . . . . . . . 13 ⊢ (𝑥 = {0, 2} → (𝑥 ≈ 2o ↔ {0, 2} ≈ 2o))
120	118, 119	orbi12d 805	. . . . . . . . . . . 12 ⊢ (𝑥 = {0, 2} → ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) ↔ ({0, 2} ≈ 1o ∨ {0, 2} ≈ 2o)))
121	120	elrab 2982	. . . . . . . . . . 11 ⊢ ({0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ↔ ({0, 2} ∈ 𝒫 (0...3) ∧ ({0, 2} ≈ 1o ∨ {0, 2} ≈ 2o)))
122	112, 117, 121	mpbir2an 955	. . . . . . . . . 10 ⊢ {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
123	122	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {0, 2} ∈ {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
124	103, 123	s2cld 11528	. . . . . . . 8 ⊢ (⊤ → ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
125	90	elexi 2834	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1}”⟩ ∈ V
126	5, 125	opvtxfvi 16182	. . . . . . . . . . 11 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
127	126	eqcomi 2242	. . . . . . . . . 10 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
128	5, 125	opiedgfvi 16183	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
129	128	eqcomi 2242	. . . . . . . . . 10 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
130	96	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
131		0ex 4255	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
132	5, 131	opvtxfvi 16182	. . . . . . . . . . . . 13 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
133	132	eqcomi 2242	. . . . . . . . . . . 12 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
134	5, 131	opiedgfvi 16183	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
135	134	eqcomi 2242	. . . . . . . . . . . 12 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
136		wrd0 11307	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
137	136	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → ∅ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
138		eqid 2238	. . . . . . . . . . . . . 14 ⊢ ∅ = ∅
139	138	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ = ∅)
140	4	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → (0...3) ∈ Fin)
141		upgr0eop 16277	. . . . . . . . . . . . . . 15 ⊢ ((0...3) ∈ Fin → ⟨(0...3), ∅⟩ ∈ UPGraph)
142	4, 141	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ⟨(0...3), ∅⟩ ∈ UPGraph
143	142	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → ⟨(0...3), ∅⟩ ∈ UPGraph)
144	133, 135, 28, 139, 140, 143	vtxdgfi0e 16450	. . . . . . . . . . . 12 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ∅⟩)‘0) = 0)
145	126	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3))
146		1nn0 9558	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ0
147		1le3 9495	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 3
148		elfz2nn0 10497	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
149	146, 25, 147, 148	mpbir3an 1210	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0...3)
150	149	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ∈ (0...3))
151		1ne0 9351	. . . . . . . . . . . . 13 ⊢ 1 ≠ 0
152	151	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 0)
153		ccatlid 11352	. . . . . . . . . . . . . . 15 ⊢ (⟨“{0, 1}”⟩ ∈ Word V → (∅ ++ ⟨“{0, 1}”⟩) = ⟨“{0, 1}”⟩)
154	90, 153	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (∅ ++ ⟨“{0, 1}”⟩) = ⟨“{0, 1}”⟩
155	128, 154	eqtr4i 2262	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
156	155	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩))
157	133, 28, 135, 137, 144, 145, 140, 150, 152, 156	vdegp1bid 16470	. . . . . . . . . . 11 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = (0 + 1))
158		0p1e1 9397	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
159	157, 158	eqtrdi 2287	. . . . . . . . . 10 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘0) = 1)
160	82	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3))
161	110	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → 2 ∈ (0...3))
162		2ne0 9375	. . . . . . . . . . 11 ⊢ 2 ≠ 0
163	162	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → 2 ≠ 0)
164		df-s2 11506	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
165	84, 164	eqtri 2259	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
166	165	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩))
167	127, 28, 129, 130, 159, 160, 140, 161, 163, 166	vdegp1bid 16470	. . . . . . . . 9 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = (1 + 1))
168		1p1e2 9400	. . . . . . . . 9 ⊢ (1 + 1) = 2
169	167, 168	eqtrdi 2287	. . . . . . . 8 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘0) = 2)
170	68	a1i 9	. . . . . . . 8 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3))
171		nn0fz0 10504	. . . . . . . . . 10 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
172	25, 171	mpbi 145	. . . . . . . . 9 ⊢ 3 ∈ (0...3)
173	172	a1i 9	. . . . . . . 8 ⊢ (⊤ → 3 ∈ (0...3))
174		3ne0 9378	. . . . . . . . 9 ⊢ 3 ≠ 0
175	174	a1i 9	. . . . . . . 8 ⊢ (⊤ → 3 ≠ 0)
176		df-s3 11507	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
177	70, 176	eqtri 2259	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
178	177	a1i 9	. . . . . . . 8 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩))
179	83, 28, 85, 124, 169, 170, 140, 173, 175, 178	vdegp1bid 16470	. . . . . . 7 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = (2 + 1))
180		2p1e3 9417	. . . . . . 7 ⊢ (2 + 1) = 3
181	179, 180	eqtrdi 2287	. . . . . 6 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘0) = 3)
182	54	a1i 9	. . . . . 6 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3))
183		1ne2 9490	. . . . . . 7 ⊢ 1 ≠ 2
184	183	a1i 9	. . . . . 6 ⊢ (⊤ → 1 ≠ 2)
185		df-s4 11508	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
186	56, 185	eqtri 2259	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
187	186	a1i 9	. . . . . 6 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩))
188	69, 28, 71, 78, 181, 182, 140, 150, 152, 161, 163, 184, 187	vdegp1aid 16469	. . . . 5 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘0) = 3)
189	43	a1i 9	. . . . 5 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3))
190		df-s5 11509	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
191	45, 190	eqtri 2259	. . . . . 6 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
192	191	a1i 9	. . . . 5 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩))
193	55, 28, 57, 61, 188, 189, 140, 150, 152, 161, 163, 184, 192	vdegp1aid 16469	. . . 4 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘0) = 3)
194	23	a1i 9	. . . 4 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3))
195	105, 107	ltneii 8412	. . . . 5 ⊢ 2 ≠ 3
196	195	a1i 9	. . . 4 ⊢ (⊤ → 2 ≠ 3)
197		df-s6 11510	. . . . . 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}”⟩)
198	29, 197	eqtri 2259	. . . . 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}”⟩)
199	198	a1i 9	. . . 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}”⟩))
200	44, 28, 46, 50, 193, 194, 140, 161, 163, 173, 175, 196, 199	vdegp1aid 16469	. . 3 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘0) = 3)
201		konigsberg.v	. . . . 5 ⊢ 𝑉 = (0...3)
202		konigsberg.e	. . . . 5 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
203		konigsberg.g	. . . . 5 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩
204	201, 202, 203	konigsbergvtx 16637	. . . 4 ⊢ (Vtx‘𝐺) = (0...3)
205	204	a1i 9	. . 3 ⊢ (⊤ → (Vtx‘𝐺) = (0...3))
206	201, 202, 203	konigsbergiedg 16638	. . . . 5 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
207	206, 33	eqtri 2259	. . . 4 ⊢ (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
208	207	a1i 9	. . 3 ⊢ (⊤ → (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩))
209	24, 28, 30, 39, 200, 205, 140, 161, 163, 173, 175, 196, 208	vdegp1aid 16469	. 2 ⊢ (⊤ → ((VtxDeg‘𝐺)‘0) = 3)
210	209	mptru 1411	1 ⊢ ((VtxDeg‘𝐺)‘0) = 3

Syntax hints:   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ⊤wtru 1403   ∈ wcel 2209   ≠ wne 2420  {crab 2532  Vcvv 2821  ∅c0 3520  𝒫 cpw 3685  {cpr 3706  ⟨cop 3708   class class class wbr 4125  ‘cfv 5372  (class class class)co 6075  1_oc1o 6670  2_oc2o 6671   ≈ cen 7010  Fincfn 7012  0cc0 8169  1c1 8170   + caddc 8172   ≤ cle 8351  ℕcn 9283  2c2 9334  3c3 9335  ℕ₀cn0 9542  ℤcz 9623  ...cfz 10390  ♯chash 11192  Word cword 11282   ++ cconcat 11336  ⟨“cs1 11361  ⟨“cs2 11499  ⟨“cs3 11500  ⟨“cs4 11501  ⟨“cs5 11502  ⟨“cs6 11503  ⟨“cs7 11504  Vtxcvtx 16167  iEdgciedg 16168  UPGraphcupgr 16246  VtxDegcvtxdg 16441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-xadd 10154  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-concat 11337  df-s1 11362  df-s2 11506  df-s3 11507  df-s4 11508  df-s5 11509  df-s6 11510  df-s7 11511  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-upgren 16248  df-umgren 16249  df-vtxdg 16442

This theorem is referenced by:  konigsberglem4  16646