usgrexmpl2nb3 - Mathbox for Alexander van der Vekens

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Theorem usgrexmpl2nb3 47769

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 12371	. . . . . 6 ⊢ 3 ∈ V
2	1	tpid1 4793	. . . . 5 ⊢ 3 ∈ {3, 4, 5}
3	2	olci 865	. . . 4 ⊢ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5})
4		elun 4170	. . . 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 47765	. . 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 11280	. . . . . 6 ⊢ 0 ∈ V
12	11	tpid1 4793	. . . . 5 ⊢ 0 ∈ {0, 1, 2}
13	12	orci 864	. . . 4 ⊢ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5})
14		elun 4170	. . . 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 12366	. . . . . 6 ⊢ 2 ∈ V
17	16	tpid3 4798	. . . . 5 ⊢ 2 ∈ {0, 1, 2}
18	17	orci 864	. . . 4 ⊢ (2 ∈ {0, 1, 2} ∨ 2 ∈ {3, 4, 5})
19		elun 4170	. . . 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 12568	. . . . . . 7 ⊢ 4 ∈ ℕ₀
22	21	elexi 3506	. . . . . 6 ⊢ 4 ∈ V
23	22	tpid2 4795	. . . . 5 ⊢ 4 ∈ {3, 4, 5}
24	23	olci 865	. . . 4 ⊢ (4 ∈ {0, 1, 2} ∨ 4 ∈ {3, 4, 5})
25		elun 4170	. . . 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 4863	. . . 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 1090	. . . . . 6 ⊢ ((𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4) ↔ (𝑛 = 0 ∨ (𝑛 = 2 ∨ 𝑛 = 4)))
29		vex 3486	. . . . . . 7 ⊢ 𝑛 ∈ V
30	29	eltp 4712	. . . . . 6 ⊢ (𝑛 ∈ {0, 2, 4} ↔ (𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4))
31		prex 5455	. . . . . . . 8 ⊢ {3, 𝑛} ∈ V
32		el7g 4713	. . . . . . . 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 4757	. . . . . . . . . 10 ⊢ {0, 3} = {3, 0}
35	34	eqeq2i 2747	. . . . . . . . 9 ⊢ ({3, 𝑛} = {0, 3} ↔ {3, 𝑛} = {3, 0})
36	29	a1i 11	. . . . . . . . . . 11 ⊢ (0 ∈ V → 𝑛 ∈ V)
37		elex 3504	. . . . . . . . . . 11 ⊢ (0 ∈ V → 0 ∈ V)
38	36, 37	preq2b 4872	. . . . . . . . . 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 1092	. . . . . . . . . 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 11286	. . . . . . . . . . . . . . . 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 12462	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
47	43, 46	gtneii 11398	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 1
48		2re 12363	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
49		2lt3 12461	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 3
50	48, 49	gtneii 11398	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 2
51	47, 50	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (3 ≠ 1 ∧ 3 ≠ 2)
52	51	orci 864	. . . . . . . . . . . . . 14 ⊢ ((3 ≠ 1 ∧ 3 ≠ 2) ∨ (𝑛 ≠ 1 ∧ 𝑛 ≠ 2))
53		prneimg 4879	. . . . . . . . . . . . . 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 2944	. . . . . . . . . . . 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 12395	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
60	59, 47	pm3.2i 470	. . . . . . . . . . . . . . . . 17 ⊢ (3 ≠ 0 ∧ 3 ≠ 1)
61	60	orci 864	. . . . . . . . . . . . . . . 16 ⊢ ((3 ≠ 0 ∧ 3 ≠ 1) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 1))
62		prneimg 4879	. . . . . . . . . . . . . . . 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 2944	. . . . . . . . . . . . . 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 1094	. . . . . . . . . . 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 4757	. . . . . . . . . . . 12 ⊢ {2, 3} = {3, 2}
71	70	eqeq2i 2747	. . . . . . . . . . 11 ⊢ ({3, 𝑛} = {2, 3} ↔ {3, 𝑛} = {3, 2})
72	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 𝑛 ∈ V)
73		elex 3504	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 2 ∈ V)
74	72, 73	preq2b 4872	. . . . . . . . . . . 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 1092	. . . . . . . . . 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 12569	. . . . . . . . . . . . . . 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 12369	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
83		3lt4 12463	. . . . . . . . . . . . . . . 16 ⊢ 3 < 4
84	82, 83	ltneii 11399	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 4
85		3lt5 12467	. . . . . . . . . . . . . . . 16 ⊢ 3 < 5
86	82, 85	ltneii 11399	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 5
87	84, 86	pm3.2i 470	. . . . . . . . . . . . . 14 ⊢ (3 ≠ 4 ∧ 3 ≠ 5)
88	87	orci 864	. . . . . . . . . . . . 13 ⊢ ((3 ≠ 4 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 4 ∧ 𝑛 ≠ 5))
89		prneimg 4879	. . . . . . . . . . . . 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 2944	. . . . . . . . . . 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 864	. . . . . . . . . . . . . . 15 ⊢ ((3 ≠ 0 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 5))
97		prneimg 4879	. . . . . . . . . . . . . . 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 2944	. . . . . . . . . . . . 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 3504	. . . . . . . . . . . 12 ⊢ (4 ∈ ℕ₀ → 4 ∈ V)
105	103, 104	preq2b 4872	. . . . . . . . . . 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 913	. . . . . . . 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 913	. . . . . . 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 4103	. . 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 2765	1 ⊢ (𝐺 NeighbVtx 3) = {0, 2, 4}

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∨ w3o 1086 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 {crab 3438 Vcvv 3482 ∪ cun 3968 {csn 4648 {cpr 4650 {ctp 4652 ⟨cop 4654 (class class class)co 7445 ℝcr 11179 0cc0 11180 1c1 11181 2c2 12344 3c3 12345 4c4 12346 5c5 12347 ℕ₀cn0 12549 ...cfz 13563 ⟨“cs7 14891 NeighbVtx cnbgr 29358

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 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-dju 9966 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-n0 12550 df-xnn0 12622 df-z 12636 df-uz 12900 df-fz 13564 df-fzo 13708 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-s2 14893 df-s3 14894 df-s4 14895 df-s5 14896 df-s6 14897 df-s7 14898 df-vtx 29024 df-iedg 29025 df-edg 29074 df-upgr 29108 df-umgr 29109 df-usgr 29177 df-nbgr 29359

This theorem is referenced by: usgrexmpl2trifr 47772

W3C validator