usgrexmpl2nb3 - Mathbox for Alexander van der Vekens

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Theorem usgrexmpl2nb3 47942

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 12352	. . . . . 6 ⊢ 3 ∈ V
2	1	tpid1 4774	. . . . 5 ⊢ 3 ∈ {3, 4, 5}
3	2	olci 866	. . . 4 ⊢ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5})
4		elun 4164	. . . 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 47938	. . 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 11259	. . . . . 6 ⊢ 0 ∈ V
12	11	tpid1 4774	. . . . 5 ⊢ 0 ∈ {0, 1, 2}
13	12	orci 865	. . . 4 ⊢ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5})
14		elun 4164	. . . 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 12347	. . . . . 6 ⊢ 2 ∈ V
17	16	tpid3 4779	. . . . 5 ⊢ 2 ∈ {0, 1, 2}
18	17	orci 865	. . . 4 ⊢ (2 ∈ {0, 1, 2} ∨ 2 ∈ {3, 4, 5})
19		elun 4164	. . . 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 12549	. . . . . . 7 ⊢ 4 ∈ ℕ₀
22	21	elexi 3502	. . . . . 6 ⊢ 4 ∈ V
23	22	tpid2 4776	. . . . 5 ⊢ 4 ∈ {3, 4, 5}
24	23	olci 866	. . . 4 ⊢ (4 ∈ {0, 1, 2} ∨ 4 ∈ {3, 4, 5})
25		elun 4164	. . . 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 4844	. . . 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 3483	. . . . . . 7 ⊢ 𝑛 ∈ V
30	29	eltp 4695	. . . . . 6 ⊢ (𝑛 ∈ {0, 2, 4} ↔ (𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4))
31		prex 5444	. . . . . . . 8 ⊢ {3, 𝑛} ∈ V
32		el7g 4696	. . . . . . . 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 4738	. . . . . . . . . 10 ⊢ {0, 3} = {3, 0}
35	34	eqeq2i 2749	. . . . . . . . 9 ⊢ ({3, 𝑛} = {0, 3} ↔ {3, 𝑛} = {3, 0})
36	29	a1i 11	. . . . . . . . . . 11 ⊢ (0 ∈ V → 𝑛 ∈ V)
37		elex 3500	. . . . . . . . . . 11 ⊢ (0 ∈ V → 0 ∈ V)
38	36, 37	preq2b 4853	. . . . . . . . . 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 11265	. . . . . . . . . . . . . . . 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 12443	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
47	43, 46	gtneii 11377	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 1
48		2re 12344	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
49		2lt3 12442	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 3
50	48, 49	gtneii 11377	. . . . . . . . . . . . . . . 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 4860	. . . . . . . . . . . . . 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 2941	. . . . . . . . . . . 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 12376	. . . . . . . . . . . . . . . . . 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 4860	. . . . . . . . . . . . . . . 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 2941	. . . . . . . . . . . . . 14 ⊢ ¬ {3, 𝑛} = {0, 1}
65	64	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ {3, 𝑛} = {1, 2} → ¬ {3, 𝑛} = {0, 1})
66	56, 65	3bior2fd 1477	. . . . . . . . . . . 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 4738	. . . . . . . . . . . 12 ⊢ {2, 3} = {3, 2}
71	70	eqeq2i 2749	. . . . . . . . . . 11 ⊢ ({3, 𝑛} = {2, 3} ↔ {3, 𝑛} = {3, 2})
72	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 𝑛 ∈ V)
73		elex 3500	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 2 ∈ V)
74	72, 73	preq2b 4853	. . . . . . . . . . . 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 12550	. . . . . . . . . . . . . . 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 12350	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
83		3lt4 12444	. . . . . . . . . . . . . . . 16 ⊢ 3 < 4
84	82, 83	ltneii 11378	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 4
85		3lt5 12448	. . . . . . . . . . . . . . . 16 ⊢ 3 < 5
86	82, 85	ltneii 11378	. . . . . . . . . . . . . . 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 4860	. . . . . . . . . . . . 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 2941	. . . . . . . . . . 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 4860	. . . . . . . . . . . . . . 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 2941	. . . . . . . . . . . . 13 ⊢ ¬ {3, 𝑛} = {0, 5}
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ {3, 𝑛} = {4, 5} → ¬ {3, 𝑛} = {0, 5})
101	92, 100	3bior2fd 1477	. . . . . . . . . . 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 3500	. . . . . . . . . . . 12 ⊢ (4 ∈ ℕ₀ → 4 ∈ V)
105	103, 104	preq2b 4853	. . . . . . . . . . 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 4097	. . 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 1461	. 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 2767	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 1538 ∈ wcel 2107 ≠ wne 2939 {crab 3434 Vcvv 3479 ∪ cun 3962 {csn 4632 {cpr 4634 {ctp 4636 ⟨cop 4638 (class class class)co 7435 ℝcr 11158 0cc0 11159 1c1 11160 2c2 12325 3c3 12326 4c4 12327 5c5 12328 ℕ₀cn0 12530 ...cfz 13550 ⟨“cs7 14888 NeighbVtx cnbgr 29372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5286 ax-sep 5303 ax-nul 5313 ax-pow 5372 ax-pr 5439 ax-un 7758 ax-cnex 11215 ax-resscn 11216 ax-1cn 11217 ax-icn 11218 ax-addcl 11219 ax-addrcl 11220 ax-mulcl 11221 ax-mulrcl 11222 ax-mulcom 11223 ax-addass 11224 ax-mulass 11225 ax-distr 11226 ax-i2m1 11227 ax-1ne0 11228 ax-1rid 11229 ax-rnegex 11230 ax-rrecex 11231 ax-cnre 11232 ax-pre-lttri 11233 ax-pre-lttrn 11234 ax-pre-ltadd 11235 ax-pre-mulgt0 11236

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1541 df-fal 1551 df-ex 1778 df-nf 1782 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3435 df-v 3481 df-sbc 3793 df-csb 3910 df-dif 3967 df-un 3969 df-in 3971 df-ss 3981 df-pss 3984 df-nul 4341 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4914 df-int 4953 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5642 df-we 5644 df-xp 5696 df-rel 5697 df-cnv 5698 df-co 5699 df-dm 5700 df-rn 5701 df-res 5702 df-ima 5703 df-pred 6326 df-ord 6392 df-on 6393 df-lim 6394 df-suc 6395 df-iota 6519 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7392 df-ov 7438 df-oprab 7439 df-mpo 7440 df-om 7892 df-1st 8019 df-2nd 8020 df-frecs 8311 df-wrecs 8342 df-recs 8416 df-rdg 8455 df-1o 8511 df-2o 8512 df-oadd 8515 df-er 8750 df-en 8991 df-dom 8992 df-sdom 8993 df-fin 8994 df-dju 9945 df-card 9983 df-pnf 11301 df-mnf 11302 df-xr 11303 df-ltxr 11304 df-le 11305 df-sub 11498 df-neg 11499 df-nn 12271 df-2 12333 df-3 12334 df-4 12335 df-5 12336 df-6 12337 df-7 12338 df-n0 12531 df-xnn0 12604 df-z 12618 df-uz 12883 df-fz 13551 df-fzo 13698 df-hash 14373 df-word 14556 df-concat 14612 df-s1 14637 df-s2 14890 df-s3 14891 df-s4 14892 df-s5 14893 df-s6 14894 df-s7 14895 df-vtx 29038 df-iedg 29039 df-edg 29088 df-upgr 29122 df-umgr 29123 df-usgr 29191 df-nbgr 29373

This theorem is referenced by: usgrexmpl2trifr 47945

W3C validator