usgrexmpl2nb3 - Mathbox for Alexander van der Vekens

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Theorem usgrexmpl2nb3 48526

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 12261	. . . . . 6 ⊢ 3 ∈ V
2	1	tpid1 4707	. . . . 5 ⊢ 3 ∈ {3, 4, 5}
3	2	olci 872	. . . 4 ⊢ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5})
4		elun 4090	. . . 4 ⊢ (3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5}))
5	3, 4	mpbir 232	. . 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 48522	. . 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 11136	. . . . . 6 ⊢ 0 ∈ V
12	11	tpid1 4707	. . . . 5 ⊢ 0 ∈ {0, 1, 2}
13	12	orci 871	. . . 4 ⊢ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5})
14		elun 4090	. . . 4 ⊢ (0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5}))
15	13, 14	mpbir 232	. . 3 ⊢ 0 ∈ ({0, 1, 2} ∪ {3, 4, 5})
16		2ex 12256	. . . . . 6 ⊢ 2 ∈ V
17	16	tpid3 4712	. . . . 5 ⊢ 2 ∈ {0, 1, 2}
18	17	orci 871	. . . 4 ⊢ (2 ∈ {0, 1, 2} ∨ 2 ∈ {3, 4, 5})
19		elun 4090	. . . 4 ⊢ (2 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (2 ∈ {0, 1, 2} ∨ 2 ∈ {3, 4, 5}))
20	18, 19	mpbir 232	. . 3 ⊢ 2 ∈ ({0, 1, 2} ∪ {3, 4, 5})
21		4nn0 12454	. . . . . . 7 ⊢ 4 ∈ ℕ₀
22	21	elexi 3455	. . . . . 6 ⊢ 4 ∈ V
23	22	tpid2 4709	. . . . 5 ⊢ 4 ∈ {3, 4, 5}
24	23	olci 872	. . . 4 ⊢ (4 ∈ {0, 1, 2} ∨ 4 ∈ {3, 4, 5})
25		elun 4090	. . . 4 ⊢ (4 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (4 ∈ {0, 1, 2} ∨ 4 ∈ {3, 4, 5}))
26	24, 25	mpbir 232	. . 3 ⊢ 4 ∈ ({0, 1, 2} ∪ {3, 4, 5})
27		tpssi 4776	. . . 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 1095	. . . . . 6 ⊢ ((𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4) ↔ (𝑛 = 0 ∨ (𝑛 = 2 ∨ 𝑛 = 4)))
29		vex 3436	. . . . . . 7 ⊢ 𝑛 ∈ V
30	29	eltp 4628	. . . . . 6 ⊢ (𝑛 ∈ {0, 2, 4} ↔ (𝑛 = 0 ∨ 𝑛 = 2 ∨ 𝑛 = 4))
31		prex 5374	. . . . . . . 8 ⊢ {3, 𝑛} ∈ V
32		el7g 4629	. . . . . . . 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 4671	. . . . . . . . . 10 ⊢ {0, 3} = {3, 0}
35	34	eqeq2i 2753	. . . . . . . . 9 ⊢ ({3, 𝑛} = {0, 3} ↔ {3, 𝑛} = {3, 0})
36	29	a1i 11	. . . . . . . . . . 11 ⊢ (0 ∈ V → 𝑛 ∈ V)
37		elex 3453	. . . . . . . . . . 11 ⊢ (0 ∈ V → 0 ∈ V)
38	36, 37	preq2b 4785	. . . . . . . . . 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 276	. . . . . . . 8 ⊢ ({3, 𝑛} = {0, 3} ↔ 𝑛 = 0)
41		3orrot 1097	. . . . . . . . . 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 471	. . . . . . . . . . . . . . 15 ⊢ (3 ∈ V ∧ 𝑛 ∈ V)
43		1re 11142	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
44	43, 16	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ ∧ 2 ∈ V)
45	42, 44	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (1 ∈ ℝ ∧ 2 ∈ V))
46		1lt3 12347	. . . . . . . . . . . . . . . . 17 ⊢ 1 < 3
47	43, 46	gtneii 11256	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 1
48		2re 12253	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
49		2lt3 12346	. . . . . . . . . . . . . . . . 17 ⊢ 2 < 3
50	48, 49	gtneii 11256	. . . . . . . . . . . . . . . 16 ⊢ 3 ≠ 2
51	47, 50	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ (3 ≠ 1 ∧ 3 ≠ 2)
52	51	orci 871	. . . . . . . . . . . . . 14 ⊢ ((3 ≠ 1 ∧ 3 ≠ 2) ∨ (𝑛 ≠ 1 ∧ 𝑛 ≠ 2))
53		prneimg 4792	. . . . . . . . . . . . . 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 2937	. . . . . . . . . . . 12 ⊢ ¬ {3, 𝑛} = {1, 2}
56		id 22	. . . . . . . . . . . . 13 ⊢ (¬ {3, 𝑛} = {1, 2} → ¬ {3, 𝑛} = {1, 2})
57	11, 43	pm3.2i 471	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V ∧ 1 ∈ ℝ)
58	42, 57	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (0 ∈ V ∧ 1 ∈ ℝ))
59		3ne0 12285	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
60	59, 47	pm3.2i 471	. . . . . . . . . . . . . . . . 17 ⊢ (3 ≠ 0 ∧ 3 ≠ 1)
61	60	orci 871	. . . . . . . . . . . . . . . 16 ⊢ ((3 ≠ 0 ∧ 3 ≠ 1) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 1))
62		prneimg 4792	. . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . 14 ⊢ ¬ {3, 𝑛} = {0, 1}
65	64	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ {3, 𝑛} = {1, 2} → ¬ {3, 𝑛} = {0, 1})
66	56, 65	3bior2fd 1485	. . . . . . . . . . . 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 1099	. . . . . . . . . . 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 276	. . . . . . . . . 10 ⊢ ({3, 𝑛} = {2, 3} ↔ ({3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3} ∨ {3, 𝑛} = {0, 1}))
70		prcom 4671	. . . . . . . . . . . 12 ⊢ {2, 3} = {3, 2}
71	70	eqeq2i 2753	. . . . . . . . . . 11 ⊢ ({3, 𝑛} = {2, 3} ↔ {3, 𝑛} = {3, 2})
72	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 𝑛 ∈ V)
73		elex 3453	. . . . . . . . . . . . 13 ⊢ (2 ∈ V → 2 ∈ V)
74	72, 73	preq2b 4785	. . . . . . . . . . . 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 276	. . . . . . . . . 10 ⊢ ({3, 𝑛} = {2, 3} ↔ 𝑛 = 2)
77	41, 69, 76	3bitr2i 300	. . . . . . . . 9 ⊢ (({3, 𝑛} = {0, 1} ∨ {3, 𝑛} = {1, 2} ∨ {3, 𝑛} = {2, 3}) ↔ 𝑛 = 2)
78		3orrot 1097	. . . . . . . . . 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 12455	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ ℕ₀
80	21, 79	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀)
81	42, 80	pm3.2i 471	. . . . . . . . . . . . 13 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀))
82		3re 12259	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
83		3lt4 12348	. . . . . . . . . . . . . . . 16 ⊢ 3 < 4
84	82, 83	ltneii 11257	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 4
85		3lt5 12352	. . . . . . . . . . . . . . . 16 ⊢ 3 < 5
86	82, 85	ltneii 11257	. . . . . . . . . . . . . . 15 ⊢ 3 ≠ 5
87	84, 86	pm3.2i 471	. . . . . . . . . . . . . 14 ⊢ (3 ≠ 4 ∧ 3 ≠ 5)
88	87	orci 871	. . . . . . . . . . . . 13 ⊢ ((3 ≠ 4 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 4 ∧ 𝑛 ≠ 5))
89		prneimg 4792	. . . . . . . . . . . . 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 2937	. . . . . . . . . . 11 ⊢ ¬ {3, 𝑛} = {4, 5}
92		id 22	. . . . . . . . . . . 12 ⊢ (¬ {3, 𝑛} = {4, 5} → ¬ {3, 𝑛} = {4, 5})
93	11, 79	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ V ∧ 5 ∈ ℕ₀)
94	42, 93	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ V ∧ 𝑛 ∈ V) ∧ (0 ∈ V ∧ 5 ∈ ℕ₀))
95	59, 86	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ (3 ≠ 0 ∧ 3 ≠ 5)
96	95	orci 871	. . . . . . . . . . . . . . 15 ⊢ ((3 ≠ 0 ∧ 3 ≠ 5) ∨ (𝑛 ≠ 0 ∧ 𝑛 ≠ 5))
97		prneimg 4792	. . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . 13 ⊢ ¬ {3, 𝑛} = {0, 5}
100	99	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ {3, 𝑛} = {4, 5} → ¬ {3, 𝑛} = {0, 5})
101	92, 100	3bior2fd 1485	. . . . . . . . . . 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 3453	. . . . . . . . . . . 12 ⊢ (4 ∈ ℕ₀ → 4 ∈ V)
105	103, 104	preq2b 4785	. . . . . . . . . . 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 300	. . . . . . . . 9 ⊢ (({3, 𝑛} = {3, 4} ∨ {3, 𝑛} = {4, 5} ∨ {3, 𝑛} = {0, 5}) ↔ 𝑛 = 4)
108	77, 107	orbi12i 920	. . . . . . . 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 920	. . . . . . 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 276	. . . . . 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 304	. . . . 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 4024	. . 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 1469	. 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 2766	1 ⊢ (𝐺 NeighbVtx 3) = {0, 2, 4}

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 853 ∨ w3o 1091 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 {crab 3392 Vcvv 3432 ∪ cun 3888 {csn 4562 {cpr 4564 {ctp 4566 ⟨cop 4568 (class class class)co 7363 ℝcr 11035 0cc0 11036 1c1 11037 2c2 12234 3c3 12235 4c4 12236 5c5 12237 ℕ₀cn0 12435 ...cfz 13459 ⟨“cs7 14806 NeighbVtx cnbgr 29426

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-oadd 8406 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9823 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-n0 12436 df-xnn0 12509 df-z 12523 df-uz 12787 df-fz 13460 df-fzo 13607 df-hash 14291 df-word 14474 df-concat 14531 df-s1 14557 df-s2 14808 df-s3 14809 df-s4 14810 df-s5 14811 df-s6 14812 df-s7 14813 df-vtx 29092 df-iedg 29093 df-edg 29142 df-upgr 29176 df-umgr 29177 df-usgr 29245 df-nbgr 29427

This theorem is referenced by: usgrexmpl2trifr 48529

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