usgrexmpl2nb3 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > usgrexmpl2nb3		Structured version Visualization version GIF version

Theorem usgrexmpl2nb3 48025

Description: The neighborhood of the forth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)

**Hypotheses**
Ref	Expression
usgrexmpl2.v	⊢ 𝑉 = (0...5)
usgrexmpl2.e	⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩
usgrexmpl2.g	⊢ 𝐺 = ⟨𝑉, 𝐸⟩

**Assertion**
Ref	Expression
usgrexmpl2nb3	⊢ (𝐺 NeighbVtx 3) = {0, 2, 4}

Proof of Theorem usgrexmpl2nb3 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3ex 12268	. . . . . 6 ⊢ 3 ∈ V
2	1	tpid1 4732	. . . . 5 ⊢ 3 ∈ {3, 4, 5}
3	2	olci 866	. . . 4 ⊢ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5})
4		elun 4116	. . . 4 ⊢ (3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5}))
5	3, 4	mpbir 231	. . 3 ⊢ 3 ∈ ({0, 1, 2} ∪ {3, 4, 5})
6		usgrexmpl2.v	. . . 4 ⊢ 𝑉 = (0...5)
7		usgrexmpl2.e	. . . 4 ⊢ 𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩
8		usgrexmpl2.g	. . . 4 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩
9	6, 7, 8	usgrexmpl2nblem 48021	. . 3 ⊢ (3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) → (𝐺 NeighbVtx 3) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))})
10	5, 9	ax-mp 5	. 2 ⊢ (𝐺 NeighbVtx 3) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))}
11		c0ex 11168	. . . . . 6 ⊢ 0 ∈ V
12	11	tpid1 4732	. . . . 5 ⊢ 0 ∈ {0, 1, 2}
13	12	orci 865	. . . 4 ⊢ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5})
14		elun 4116	. . . 4 ⊢ (0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5}))
15	13, 14	mpbir 231	. . 3 ⊢ 0 ∈ ({0, 1, 2} ∪ {3, 4, 5})
16		2ex 12263	. . . . . 6 ⊢ 2 ∈ V
17	16	tpid3 4737	. . . . 5 ⊢ 2 ∈ {0, 1, 2}
18	17	orci 865	. . . 4 ⊢ (2 ∈ {0, 1, 2} ∨ 2 ∈ {3, 4, 5})
19		elun 4116	. . . 4 ⊢ (2 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (2 ∈ {0, 1, 2} ∨ 2 ∈ {3, 4, 5}))
20	18, 19	mpbir 231	. . 3 ⊢ 2 ∈ ({0, 1, 2} ∪ {3, 4, 5})
21		4nn0 12461	. . . . . . 7 ⊢ 4 ∈ ℕ₀
22	21	elexi 3470	. . . . . 6 ⊢ 4 ∈ V
23	22	tpid2 4734	. . . . 5 ⊢ 4 ∈ {3, 4, 5}
24	23	olci 866	. . . 4 ⊢ (4 ∈ {0, 1, 2} ∨ 4 ∈ {3, 4, 5})
25		elun 4116	. . . 4 ⊢ (4 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (4 ∈ {0, 1, 2} ∨ 4 ∈ {3, 4, 5}))
26	24, 25	mpbir 231	. . 3 ⊢ 4 ∈ ({0, 1, 2} ∪ {3, 4, 5})
27		tpssi 4802	. . . 4 ⊢ ((0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 2 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 4 ∈ ({0, 1, 2} ∪ {3, 4, 5})) → {0, 2, 4} ⊆ ({0, 1, 2} ∪ {3, 4, 5}))
28		3orass 1089	. . . . . 6 ⊢ ((𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4) ↔ (𝑛 = 0 ∨ (𝑛 = 2 ∨ 𝑛 = 4)))
29		vex 3451	. . . . . . 7 ⊢ 𝑛 ∈ V
30	29	eltp 4653	. . . . . 6 ⊢ (𝑛 ∈ {0, 2, 4} ↔ (𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4))
31		prex 5392	. . . . . . . 8 ⊢ {3, 𝑛} ∈ V
32		el7g 4654	. . . . . . . 8 ⊢ ({3, 𝑛} ∈ V → ({3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ ({3, 𝑛} = {0, 3} ∨ (({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ∨ ({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5})))))
33	31, 32	ax-mp 5	. . . . . . 7 ⊢ ({3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ ({3, 𝑛} = {0, 3} ∨ (({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ∨ ({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5}))))
34		prcom 4696	. . . . . . . . . 10 ⊢ {0, 3} = {3, 0}
35	34	eqeq2i 2742	. . . . . . . . 9 ⊢ ({3, 𝑛} = {0, 3} ↔ {3, 𝑛} = {3, 0})
36	29	a1i 11	. . . . . . . . . . 11 ⊢ (0 ∈ V → 𝑛 ∈ V)
37		elex 3468	. . . . . . . . . . 11 ⊢ (0 ∈ V → 0 ∈ V)
38	36, 37	preq2b 4811	. . . . . . . . . 10 ⊢ (0 ∈ V → ({3, 𝑛} = {3, 0} ↔ 𝑛 = 0))
39	11, 38	ax-mp 5	. . . . . . . . 9 ⊢ ({3, 𝑛} = {3, 0} ↔ 𝑛 = 0)
40	35, 39	bitri 275	. . . . . . . 8 ⊢ ({3, 𝑛} = {0, 3} ↔ 𝑛 = 0)
41		3orrot 1091	. . . . . . . . . 10 ⊢ (({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ↔ ({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3} ∨ {3, 𝑛} = {0, 1}))
42	1, 29	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ V ∧ 𝑛 ∈ V)
43		1re 11174	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
44	43, 16	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ ∧ 2 ∈ V)
45	42, 44	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (1 ∈ ℝ ∧ 2 ∈ V))
46		1lt3 12354	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
47	43, 46	gtneii 11286	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 1
48		2re 12260	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
49		2lt3 12353	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 3
50	48, 49	gtneii 11286	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 2
51	47, 50	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (3 ≠ 1 ∧ 3 ≠ 2)
52	51	orci 865	. . . . . . . . . . . . . 14 ⊢ ((3 ≠ 1 ∧ 3 ≠ 2) ∨ (𝑛 ≠ 1 ∧ 𝑛 ≠ 2))
53		prneimg 4818	. . . . . . . . . . . . . 14 ⊢ (((3 ∈ V ∧ 𝑛 ∈ V) ∧ (1 ∈ ℝ ∧ 2 ∈ V)) → (((3 ≠ 1 ∧ 3 ≠ 2) ∨ (𝑛 ≠ 1 ∧ 𝑛 ≠ 2)) → {3, 𝑛} ≠ {1, 2}))
54	45, 52, 53	mp2 9	. . . . . . . . . . . . 13 ⊢ {3, 𝑛} ≠ {1, 2}
55	54	neii 2927	. . . . . . . . . . . 12 ⊢ ¬ {3, 𝑛} = {1, 2}
56		id 22	. . . . . . . . . . . . 13 ⊢ (¬ {3, 𝑛} = {1, 2} → ¬ {3, 𝑛} = {1, 2})
57	11, 43	pm3.2i 470	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V ∧ 1 ∈ ℝ)
58	42, 57	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (0 ∈ V ∧ 1 ∈ ℝ))
59		3ne0 12292	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
60	59, 47	pm3.2i 470	. . . . . . . . . . . . . . . . 17 ⊢ (3 ≠ 0 ∧ 3 ≠ 1)
61	60	orci 865	. . . . . . . . . . . . . . . 16 ⊢ ((3 ≠ 0 ∧ 3 ≠ 1) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 1))
62		prneimg 4818	. . . . . . . . . . . . . . . 16 ⊢ (((3 ∈ V ∧ 𝑛 ∈ V) ∧ (0 ∈ V ∧ 1 ∈ ℝ)) → (((3 ≠ 0 ∧ 3 ≠ 1) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 1)) → {3, 𝑛} ≠ {0, 1}))
63	58, 61, 62	mp2 9	. . . . . . . . . . . . . . 15 ⊢ {3, 𝑛} ≠ {0, 1}
64	63	neii 2927	. . . . . . . . . . . . . 14 ⊢ ¬ {3, 𝑛} = {0, 1}
65	64	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ {3, 𝑛} = {1, 2} → ¬ {3, 𝑛} = {0, 1})
66	56, 65	3bior2fd 1479	. . . . . . . . . . . 12 ⊢ (¬ {3, 𝑛} = {1, 2} → ({3, 𝑛} = {2, 3} ↔ ({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {2, 3})))
67	55, 66	ax-mp 5	. . . . . . . . . . 11 ⊢ ({3, 𝑛} = {2, 3} ↔ ({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {2, 3}))
68		3orcomb 1093	. . . . . . . . . . 11 ⊢ (({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {2, 3}) ↔ ({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3} ∨ {3, 𝑛} = {0, 1}))
69	67, 68	bitri 275	. . . . . . . . . 10 ⊢ ({3, 𝑛} = {2, 3} ↔ ({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3} ∨ {3, 𝑛} = {0, 1}))
70		prcom 4696	. . . . . . . . . . . 12 ⊢ {2, 3} = {3, 2}
71	70	eqeq2i 2742	. . . . . . . . . . 11 ⊢ ({3, 𝑛} = {2, 3} ↔ {3, 𝑛} = {3, 2})
72	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 𝑛 ∈ V)
73		elex 3468	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 2 ∈ V)
74	72, 73	preq2b 4811	. . . . . . . . . . . 12 ⊢ (2 ∈ V → ({3, 𝑛} = {3, 2} ↔ 𝑛 = 2))
75	16, 74	ax-mp 5	. . . . . . . . . . 11 ⊢ ({3, 𝑛} = {3, 2} ↔ 𝑛 = 2)
76	71, 75	bitri 275	. . . . . . . . . 10 ⊢ ({3, 𝑛} = {2, 3} ↔ 𝑛 = 2)
77	41, 69, 76	3bitr2i 299	. . . . . . . . 9 ⊢ (({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ↔ 𝑛 = 2)
78		3orrot 1091	. . . . . . . . . 10 ⊢ (({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5}) ↔ ({3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5} ∨ {3, 𝑛} = {3, 4}))
79		5nn0 12462	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ ℕ₀
80	21, 79	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀)
81	42, 80	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀))
82		3re 12266	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
83		3lt4 12355	. . . . . . . . . . . . . . . 16 ⊢ 3 < 4
84	82, 83	ltneii 11287	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 4
85		3lt5 12359	. . . . . . . . . . . . . . . 16 ⊢ 3 < 5
86	82, 85	ltneii 11287	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 5
87	84, 86	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (3 ≠ 4 ∧ 3 ≠ 5)
88	87	orci 865	. . . . . . . . . . . . 13 ⊢ ((3 ≠ 4 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 4 ∧ 𝑛 ≠ 5))
89		prneimg 4818	. . . . . . . . . . . . 13 ⊢ (((3 ∈ V ∧ 𝑛 ∈ V) ∧ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀)) → (((3 ≠ 4 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 4 ∧ 𝑛 ≠ 5)) → {3, 𝑛} ≠ {4, 5}))
90	81, 88, 89	mp2 9	. . . . . . . . . . . 12 ⊢ {3, 𝑛} ≠ {4, 5}
91	90	neii 2927	. . . . . . . . . . 11 ⊢ ¬ {3, 𝑛} = {4, 5}
92		id 22	. . . . . . . . . . . 12 ⊢ (¬ {3, 𝑛} = {4, 5} → ¬ {3, 𝑛} = {4, 5})
93	11, 79	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V ∧ 5 ∈ ℕ₀)
94	42, 93	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (0 ∈ V ∧ 5 ∈ ℕ₀))
95	59, 86	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (3 ≠ 0 ∧ 3 ≠ 5)
96	95	orci 865	. . . . . . . . . . . . . . 15 ⊢ ((3 ≠ 0 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 5))
97		prneimg 4818	. . . . . . . . . . . . . . 15 ⊢ (((3 ∈ V ∧ 𝑛 ∈ V) ∧ (0 ∈ V ∧ 5 ∈ ℕ₀)) → (((3 ≠ 0 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 5)) → {3, 𝑛} ≠ {0, 5}))
98	94, 96, 97	mp2 9	. . . . . . . . . . . . . 14 ⊢ {3, 𝑛} ≠ {0, 5}
99	98	neii 2927	. . . . . . . . . . . . 13 ⊢ ¬ {3, 𝑛} = {0, 5}
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ {3, 𝑛} = {4, 5} → ¬ {3, 𝑛} = {0, 5})
101	92, 100	3bior2fd 1479	. . . . . . . . . . 11 ⊢ (¬ {3, 𝑛} = {4, 5} → ({3, 𝑛} = {3, 4} ↔ ({3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5} ∨ {3, 𝑛} = {3, 4})))
102	91, 101	ax-mp 5	. . . . . . . . . 10 ⊢ ({3, 𝑛} = {3, 4} ↔ ({3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5} ∨ {3, 𝑛} = {3, 4}))
103	29	a1i 11	. . . . . . . . . . . 12 ⊢ (4 ∈ ℕ₀ → 𝑛 ∈ V)
104		elex 3468	. . . . . . . . . . . 12 ⊢ (4 ∈ ℕ₀ → 4 ∈ V)
105	103, 104	preq2b 4811	. . . . . . . . . . 11 ⊢ (4 ∈ ℕ₀ → ({3, 𝑛} = {3, 4} ↔ 𝑛 = 4))
106	21, 105	ax-mp 5	. . . . . . . . . 10 ⊢ ({3, 𝑛} = {3, 4} ↔ 𝑛 = 4)
107	78, 102, 106	3bitr2i 299	. . . . . . . . 9 ⊢ (({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5}) ↔ 𝑛 = 4)
108	77, 107	orbi12i 914	. . . . . . . 8 ⊢ ((({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ∨ ({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5})) ↔ (𝑛 = 2 ∨ 𝑛 = 4))
109	40, 108	orbi12i 914	. . . . . . 7 ⊢ (({3, 𝑛} = {0, 3} ∨ (({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ∨ ({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5}))) ↔ (𝑛 = 0 ∨ (𝑛 = 2 ∨ 𝑛 = 4)))
110	33, 109	bitri 275	. . . . . 6 ⊢ ({3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ (𝑛 = 0 ∨ (𝑛 = 2 ∨ 𝑛 = 4)))
111	28, 30, 110	3bitr4i 303	. . . . 5 ⊢ (𝑛 ∈ {0, 2, 4} ↔ {3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))
112	111	a1i 11	. . . 4 ⊢ (((0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 2 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 4 ∈ ({0, 1, 2} ∪ {3, 4, 5})) ∧ 𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5})) → (𝑛 ∈ {0, 2, 4} ↔ {3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
113	27, 112	eqrrabd 4049	. . 3 ⊢ ((0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 2 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 4 ∈ ({0, 1, 2} ∪ {3, 4, 5})) → {0, 2, 4} = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))})
114	15, 20, 26, 113	mp3an 1463	. 2 ⊢ {0, 2, 4} = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {3, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))}
115	10, 114	eqtr4i 2755	1 ⊢ (𝐺 NeighbVtx 3) = {0, 2, 4}

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3405 Vcvv 3447 ∪ cun 3912 {csn 4589 {cpr 4591 {ctp 4593 ⟨cop 4595 (class class class)co 7387 ℝcr 11067 0cc0 11068 1c1 11069 2c2 12241 3c3 12242 4c4 12243 5c5 12244 ℕ₀cn0 12442 ...cfz 13468 ⟨“cs7 14812 NeighbVtx cnbgr 29259

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-hash 14296 df-word 14479 df-concat 14536 df-s1 14561 df-s2 14814 df-s3 14815 df-s4 14816 df-s5 14817 df-s6 14818 df-s7 14819 df-vtx 28925 df-iedg 28926 df-edg 28975 df-upgr 29009 df-umgr 29010 df-usgr 29078 df-nbgr 29260

This theorem is referenced by: usgrexmpl2trifr 48028

W3C validator