Theorem konigsberglem3 16485

Description: Lemma 3 for konigsberg 16488: 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		0z 9588	. . . . . . . 8 ⊢ 0 ∈ ℤ
2		3z 9606	. . . . . . . 8 ⊢ 3 ∈ ℤ
3		fzfig 10792	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0...3) ∈ Fin)
4	1, 2, 3	mp2an 426	. . . . . . 7 ⊢ (0...3) ∈ Fin
5	4	elexi 2826	. . . . . 6 ⊢ (0...3) ∈ V
6		0nn0 9511	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
7		1nn0 9512	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
8		prexg 4325	. . . . . . . . . . 11 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ∈ V)
9	6, 7, 8	mp2an 426	. . . . . . . . . 10 ⊢ {0, 1} ∈ V
10	9	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {0, 1} ∈ V)
11		2nn0 9513	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ0
12		prexg 4325	. . . . . . . . . . 11 ⊢ ((0 ∈ ℕ0 ∧ 2 ∈ ℕ0) → {0, 2} ∈ V)
13	6, 11, 12	mp2an 426	. . . . . . . . . 10 ⊢ {0, 2} ∈ V
14	13	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {0, 2} ∈ V)
15		3nn0 9514	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ0
16		prexg 4325	. . . . . . . . . . 11 ⊢ ((0 ∈ ℕ0 ∧ 3 ∈ ℕ0) → {0, 3} ∈ V)
17	6, 15, 16	mp2an 426	. . . . . . . . . 10 ⊢ {0, 3} ∈ V
18	17	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {0, 3} ∈ V)
19		prexg 4325	. . . . . . . . . . 11 ⊢ ((1 ∈ ℕ0 ∧ 2 ∈ ℕ0) → {1, 2} ∈ V)
20	7, 11, 19	mp2an 426	. . . . . . . . . 10 ⊢ {1, 2} ∈ V
21	20	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {1, 2} ∈ V)
22		prexg 4325	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ0 ∧ 3 ∈ ℕ0) → {2, 3} ∈ V)
23	11, 15, 22	mp2an 426	. . . . . . . . . 10 ⊢ {2, 3} ∈ V
24	23	a1i 9	. . . . . . . . 9 ⊢ (⊤ → {2, 3} ∈ V)
25	10, 14, 18, 21, 21, 24	s6cld 11474	. . . . . . . 8 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V)
26	25	mptru 1407	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
27	26	elexi 2826	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
28	5, 27	opvtxfvi 16022	. . . . 5 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
29	28	eqcomi 2236	. . . 4 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
30		nn0fz0 10453	. . . . . 6 ⊢ (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
31	15, 30	mpbi 145	. . . . 5 ⊢ 3 ∈ (0...3)
32	31	a1i 9	. . . 4 ⊢ (⊤ → 3 ∈ (0...3))
33	5, 27	opiedgfvi 16023	. . . . 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}”⟩
34	33	eqcomi 2236	. . . 4 ⊢ ⟨“{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}”⟩⟩)
35	24	s1cld 11310	. . . . . . 7 ⊢ (⊤ → ⟨“{2, 3}”⟩ ∈ Word V)
36	35	mptru 1407	. . . . . 6 ⊢ ⟨“{2, 3}”⟩ ∈ Word V
37		df-s7 11453	. . . . . 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		eqid 2232	. . . . . . 7 ⊢ (0...3) = (0...3)
39		eqid 2232	. . . . . . 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}”⟩
40		eqid 2232	. . . . . . 7 ⊢ ⟨(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}”⟩⟩
41	38, 39, 40	konigsbergssiedgwen 16481	. . . . . 6 ⊢ ((⟨“{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)})
42	26, 36, 37, 41	mp3an 1374	. . . . 5 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
43	42	a1i 9	. . . 4 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
44	10, 14, 18, 21, 21	s5cld 11473	. . . . . . . . . 10 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V)
45	44	mptru 1407	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
46	45	elexi 2826	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
47	5, 46	opvtxfvi 16022	. . . . . . 7 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
48	47	eqcomi 2236	. . . . . 6 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
49	5, 46	opiedgfvi 16023	. . . . . . 7 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
50	49	eqcomi 2236	. . . . . 6 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
51	24, 24	s2cld 11470	. . . . . . 7 ⊢ (⊤ → ⟨“{2, 3} {2, 3}”⟩ ∈ Word V)
52	10, 14, 18, 21, 21, 24, 24	s5s2d 11497	. . . . . . 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}”⟩))
53	38, 39, 40	konigsbergssiedgwen 16481	. . . . . . 7 ⊢ ((⟨“{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)})
54	44, 51, 52, 53	syl3anc 1274	. . . . . 6 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
55	10, 14, 18, 21	s4cld 11472	. . . . . . . . . . 11 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V)
56	55	mptru 1407	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
57	56	elexi 2826	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
58	5, 57	opvtxfvi 16022	. . . . . . . 8 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
59	58	eqcomi 2236	. . . . . . 7 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
60	5, 57	opiedgfvi 16023	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
61	60	eqcomi 2236	. . . . . . 7 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
62	21, 24, 24	s3cld 11471	. . . . . . . 8 ⊢ (⊤ → ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
63	10, 14, 18, 21, 21, 24, 24	s4s3d 11494	. . . . . . . 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}”⟩))
64	38, 39, 40	konigsbergssiedgwen 16481	. . . . . . . 8 ⊢ ((⟨“{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)})
65	55, 62, 63, 64	syl3anc 1274	. . . . . . 7 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
66	10, 14, 18	s3cld 11471	. . . . . . . . . . . 12 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V)
67	66	mptru 1407	. . . . . . . . . . 11 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
68	67	elexi 2826	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
69	5, 68	opvtxfvi 16022	. . . . . . . . 9 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
70	69	eqcomi 2236	. . . . . . . 8 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
71	5, 68	opiedgfvi 16023	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
72	71	eqcomi 2236	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
73	21, 21, 24, 24	s4cld 11472	. . . . . . . . 9 ⊢ (⊤ → ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
74	10, 14, 18, 21, 21, 24, 24	s3s4d 11495	. . . . . . . . 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	38, 39, 40	konigsbergssiedgwen 16481	. . . . . . . . 9 ⊢ ((⟨“{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)})
76	66, 73, 74, 75	syl3anc 1274	. . . . . . . 8 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
77	10, 14	s2cld 11470	. . . . . . . . . . . . . 14 ⊢ (⊤ → ⟨“{0, 1} {0, 2}”⟩ ∈ Word V)
78	77	mptru 1407	. . . . . . . . . . . . 13 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
79	78	elexi 2826	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1} {0, 2}”⟩ ∈ V
80	5, 79	opvtxfvi 16022	. . . . . . . . . . 11 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
81	80	eqcomi 2236	. . . . . . . . . 10 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
82	5, 79	opiedgfvi 16023	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
83	82	eqcomi 2236	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
84	18, 21, 21, 24, 24	s5cld 11473	. . . . . . . . . . 11 ⊢ (⊤ → ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
85	10, 14, 18, 21, 21, 24, 24	s2s5d 11496	. . . . . . . . . . 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}”⟩))
86	38, 39, 40	konigsbergssiedgwen 16481	. . . . . . . . . . 11 ⊢ ((⟨“{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 ≈ 1o ∨ 𝑥 ≈ 2o)})
87	77, 84, 85, 86	syl3anc 1274	. . . . . . . . . 10 ⊢ (⊤ → ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
88	10	s1cld 11310	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ⟨“{0, 1}”⟩ ∈ Word V)
89	88	mptru 1407	. . . . . . . . . . . . . 14 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
90	89	elexi 2826	. . . . . . . . . . . . 13 ⊢ ⟨“{0, 1}”⟩ ∈ V
91	5, 90	opvtxfvi 16022	. . . . . . . . . . . 12 ⊢ (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
92	91	eqcomi 2236	. . . . . . . . . . 11 ⊢ (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
93	5, 90	opiedgfvi 16023	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
94	93	eqcomi 2236	. . . . . . . . . . 11 ⊢ ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
95	14, 18, 21, 21, 24, 24	s6cld 11474	. . . . . . . . . . . 12 ⊢ (⊤ → ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
96	10, 14, 18, 21, 21, 24, 24	s1s6d 11490	. . . . . . . . . . . 12 ⊢ (⊤ → ⟨“{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}”⟩))
97	38, 39, 40	konigsbergssiedgwen 16481	. . . . . . . . . . . 12 ⊢ ((⟨“{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)})
98	88, 95, 96, 97	syl3anc 1274	. . . . . . . . . . 11 ⊢ (⊤ → ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
99		0ex 4237	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
100	5, 99	opvtxfvi 16022	. . . . . . . . . . . . 13 ⊢ (Vtx‘⟨(0...3), ∅⟩) = (0...3)
101	100	eqcomi 2236	. . . . . . . . . . . 12 ⊢ (0...3) = (Vtx‘⟨(0...3), ∅⟩)
102	5, 99	opiedgfvi 16023	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨(0...3), ∅⟩) = ∅
103	102	eqcomi 2236	. . . . . . . . . . . 12 ⊢ ∅ = (iEdg‘⟨(0...3), ∅⟩)
104		wrd0 11249	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}
105	104	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → ∅ ∈ Word {𝑥 ∈ 𝒫 (0...3) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
106		eqidd 2233	. . . . . . . . . . . . 13 ⊢ (⊤ → ∅ = ∅)
107	4	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → (0...3) ∈ Fin)
108		upgr0eop 16117	. . . . . . . . . . . . . . 15 ⊢ ((0...3) ∈ Fin → ⟨(0...3), ∅⟩ ∈ UPGraph)
109	4, 108	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ⟨(0...3), ∅⟩ ∈ UPGraph
110	109	a1i 9	. . . . . . . . . . . . 13 ⊢ (⊤ → ⟨(0...3), ∅⟩ ∈ UPGraph)
111	101, 103, 32, 106, 107, 110	vtxdgfi0e 16290	. . . . . . . . . . . 12 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0)
112	91	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3))
113		0elfz 10452	. . . . . . . . . . . . . 14 ⊢ (3 ∈ ℕ0 → 0 ∈ (0...3))
114	15, 113	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0...3)
115	114	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ∈ (0...3))
116		3ne0 9332	. . . . . . . . . . . . . 14 ⊢ 3 ≠ 0
117	116	necomi 2497	. . . . . . . . . . . . 13 ⊢ 0 ≠ 3
118	117	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 3)
119		1le3 9449	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 3
120		elfz2nn0 10446	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
121	7, 15, 119, 120	mpbir3an 1206	. . . . . . . . . . . . 13 ⊢ 1 ∈ (0...3)
122	121	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ∈ (0...3))
123		1re 8273	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
124		1lt3 9409	. . . . . . . . . . . . . 14 ⊢ 1 < 3
125	123, 124	ltneii 8370	. . . . . . . . . . . . 13 ⊢ 1 ≠ 3
126	125	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 1 ≠ 3)
127		0ne1 9304	. . . . . . . . . . . . 13 ⊢ 0 ≠ 1
128	127	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → 0 ≠ 1)
129		s1cl 11309	. . . . . . . . . . . . . . . . 17 ⊢ ({0, 1} ∈ V → ⟨“{0, 1}”⟩ ∈ Word V)
130	9, 129	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ⟨“{0, 1}”⟩ ∈ Word V
131		ccatlid 11294	. . . . . . . . . . . . . . . 16 ⊢ (⟨“{0, 1}”⟩ ∈ Word V → (∅ ++ ⟨“{0, 1}”⟩) = ⟨“{0, 1}”⟩)
132	130, 131	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∅ ++ ⟨“{0, 1}”⟩) = ⟨“{0, 1}”⟩
133	132	eqcomi 2236	. . . . . . . . . . . . . 14 ⊢ ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
134	93, 133	eqtri 2253	. . . . . . . . . . . . 13 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
135	134	a1i 9	. . . . . . . . . . . 12 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩))
136	101, 32, 103, 105, 111, 112, 107, 115, 118, 122, 126, 128, 135	vdegp1aid 16309	. . . . . . . . . . 11 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘3) = 0)
137	80	a1i 9	. . . . . . . . . . 11 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3))
138		2re 9307	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
139		3re 9311	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℝ
140		2lt3 9408	. . . . . . . . . . . . . 14 ⊢ 2 < 3
141	138, 139, 140	ltleii 8376	. . . . . . . . . . . . 13 ⊢ 2 ≤ 3
142		elfz2nn0 10446	. . . . . . . . . . . . 13 ⊢ (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
143	11, 15, 141, 142	mpbir3an 1206	. . . . . . . . . . . 12 ⊢ 2 ∈ (0...3)
144	143	a1i 9	. . . . . . . . . . 11 ⊢ (⊤ → 2 ∈ (0...3))
145	138, 140	ltneii 8370	. . . . . . . . . . . 12 ⊢ 2 ≠ 3
146	145	a1i 9	. . . . . . . . . . 11 ⊢ (⊤ → 2 ≠ 3)
147		0ne2 9443	. . . . . . . . . . . 12 ⊢ 0 ≠ 2
148	147	a1i 9	. . . . . . . . . . 11 ⊢ (⊤ → 0 ≠ 2)
149		df-s2 11448	. . . . . . . . . . . . 13 ⊢ ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
150	82, 149	eqtri 2253	. . . . . . . . . . . 12 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
151	150	a1i 9	. . . . . . . . . . 11 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩))
152	92, 32, 94, 98, 136, 137, 107, 115, 118, 144, 146, 148, 151	vdegp1aid 16309	. . . . . . . . . 10 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘3) = 0)
153	69	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3))
154		df-s3 11449	. . . . . . . . . . . 12 ⊢ ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
155	71, 154	eqtri 2253	. . . . . . . . . . 11 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
156	155	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩))
157	81, 32, 83, 87, 152, 153, 107, 115, 118, 156	vdegp1cid 16311	. . . . . . . . 9 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = (0 + 1))
158		0p1e1 9351	. . . . . . . . 9 ⊢ (0 + 1) = 1
159	157, 158	eqtrdi 2281	. . . . . . . 8 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = 1)
160	58	a1i 9	. . . . . . . 8 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3))
161		1ne2 9444	. . . . . . . . 9 ⊢ 1 ≠ 2
162	161	a1i 9	. . . . . . . 8 ⊢ (⊤ → 1 ≠ 2)
163		df-s4 11450	. . . . . . . . . 10 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
164	60, 163	eqtri 2253	. . . . . . . . 9 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
165	164	a1i 9	. . . . . . . 8 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩))
166	70, 32, 72, 76, 159, 160, 107, 122, 126, 144, 146, 162, 165	vdegp1aid 16309	. . . . . . 7 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘3) = 1)
167	47	a1i 9	. . . . . . 7 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3))
168		df-s5 11451	. . . . . . . . 9 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
169	49, 168	eqtri 2253	. . . . . . . 8 ⊢ (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
170	169	a1i 9	. . . . . . 7 ⊢ (⊤ → (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩))
171	59, 32, 61, 65, 166, 167, 107, 122, 126, 144, 146, 162, 170	vdegp1aid 16309	. . . . . 6 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘3) = 1)
172	28	a1i 9	. . . . . 6 ⊢ (⊤ → (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3))
173		df-s6 11452	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
174	33, 173	eqtri 2253	. . . . . . 7 ⊢ (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}”⟩)
175	174	a1i 9	. . . . . 6 ⊢ (⊤ → (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}”⟩))
176	48, 32, 50, 54, 171, 172, 107, 144, 146, 175	vdegp1cid 16311	. . . . 5 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = (1 + 1))
177		1p1e2 9354	. . . . 5 ⊢ (1 + 1) = 2
178	176, 177	eqtrdi 2281	. . . 4 ⊢ (⊤ → ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = 2)
179		konigsberg.v	. . . . . 6 ⊢ 𝑉 = (0...3)
180		konigsberg.e	. . . . . 6 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
181		konigsberg.g	. . . . . 6 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩
182	179, 180, 181	konigsbergvtx 16477	. . . . 5 ⊢ (Vtx‘𝐺) = (0...3)
183	182	a1i 9	. . . 4 ⊢ (⊤ → (Vtx‘𝐺) = (0...3))
184	179, 180, 181	konigsbergiedg 16478	. . . . . 6 ⊢ (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
185	184, 37	eqtri 2253	. . . . 5 ⊢ (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
186	185	a1i 9	. . . 4 ⊢ (⊤ → (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩))
187	29, 32, 34, 43, 178, 183, 107, 144, 146, 186	vdegp1cid 16311	. . 3 ⊢ (⊤ → ((VtxDeg‘𝐺)‘3) = (2 + 1))
188	187	mptru 1407	. 2 ⊢ ((VtxDeg‘𝐺)‘3) = (2 + 1)
189		2p1e3 9371	. 2 ⊢ (2 + 1) = 3
190	188, 189	eqtri 2253	1 ⊢ ((VtxDeg‘𝐺)‘3) = 3

Syntax hints:   ∨ wo 716   = wceq 1398  ⊤wtru 1399   ∈ wcel 2203   ≠ wne 2412  {crab 2524  Vcvv 2813  ∅c0 3508  𝒫 cpw 3669  {cpr 3690  ⟨cop 3692   class class class wbr 4109  ‘cfv 5352  (class class class)co 6050  1_oc1o 6640  2_oc2o 6641   ≈ cen 6973  Fincfn 6975  0cc0 8127  1c1 8128   + caddc 8130   ≤ cle 8309  2c2 9288  3c3 9289  ℕ₀cn0 9496  ℤcz 9577  ...cfz 10342  Word cword 11224   ++ cconcat 11278  ⟨“cs1 11303  ⟨“cs2 11441  ⟨“cs3 11442  ⟨“cs4 11443  ⟨“cs5 11444  ⟨“cs6 11445  ⟨“cs7 11446  Vtxcvtx 16007  iEdgciedg 16008  UPGraphcupgr 16086  VtxDegcvtxdg 16281

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-2o 6648  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-uz 9854  df-xadd 10106  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-s1 11304  df-s2 11448  df-s3 11449  df-s4 11450  df-s5 11451  df-s6 11452  df-s7 11453  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-upgren 16088  df-umgren 16089  df-vtxdg 16282

This theorem is referenced by:  konigsberglem4  16486