usgrexmpl2nb0 - Mathbox for Alexander van der Vekens

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Theorem usgrexmpl2nb0 48763

Description: The neighborhood of the first 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
usgrexmpl2nb0	⊢ (𝐺 NeighbVtx 0) = {1, 3, 5}

Proof of Theorem usgrexmpl2nb0 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		c0ex 11199	. . . . . 6 ⊢ 0 ∈ V
2	1	tpid1 4733	. . . . 5 ⊢ 0 ∈ {0, 1, 2}
3	2	orci 878	. . . 4 ⊢ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5})
4		elun 4106	. . . 4 ⊢ (0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (0 ∈ {0, 1, 2} ∨ 0 ∈ {3, 4, 5}))
5	3, 4	mpbir 234	. . 3 ⊢ 0 ∈ ({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 48762	. . 3 ⊢ (0 ∈ ({0, 1, 2} ∪ {3, 4, 5}) → (𝐺 NeighbVtx 0) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))})
10	5, 9	ax-mp 5	. 2 ⊢ (𝐺 NeighbVtx 0) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))}
11		1eltp012 12310	. . . . 5 ⊢ 1 ∈ {0, 1, 2}
12	11	orci 878	. . . 4 ⊢ (1 ∈ {0, 1, 2} ∨ 1 ∈ {3, 4, 5})
13		elun 4106	. . . 4 ⊢ (1 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (1 ∈ {0, 1, 2} ∨ 1 ∈ {3, 4, 5}))
14	12, 13	mpbir 234	. . 3 ⊢ 1 ∈ ({0, 1, 2} ∪ {3, 4, 5})
15		3ex 12322	. . . . . 6 ⊢ 3 ∈ V
16	15	tpid1 4733	. . . . 5 ⊢ 3 ∈ {3, 4, 5}
17	16	olci 879	. . . 4 ⊢ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5})
18		elun 4106	. . . 4 ⊢ (3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (3 ∈ {0, 1, 2} ∨ 3 ∈ {3, 4, 5}))
19	17, 18	mpbir 234	. . 3 ⊢ 3 ∈ ({0, 1, 2} ∪ {3, 4, 5})
20		5nn0 12523	. . . . . . 7 ⊢ 5 ∈ ℕ₀
21	20	elexi 3475	. . . . . 6 ⊢ 5 ∈ V
22	21	tpid3 4738	. . . . 5 ⊢ 5 ∈ {3, 4, 5}
23	22	olci 879	. . . 4 ⊢ (5 ∈ {0, 1, 2} ∨ 5 ∈ {3, 4, 5})
24		elun 4106	. . . 4 ⊢ (5 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ↔ (5 ∈ {0, 1, 2} ∨ 5 ∈ {3, 4, 5}))
25	23, 24	mpbir 234	. . 3 ⊢ 5 ∈ ({0, 1, 2} ∪ {3, 4, 5})
26		tpssi 4802	. . . 4 ⊢ ((1 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 5 ∈ ({0, 1, 2} ∪ {3, 4, 5})) → {1, 3, 5} ⊆ ({0, 1, 2} ∪ {3, 4, 5}))
27		3orcoma 1107	. . . . . . 7 ⊢ ((𝑛 = 3 ∨ 𝑛 = 1 ∨ 𝑛 = 5) ↔ (𝑛 = 1 ∨ 𝑛 = 3 ∨ 𝑛 = 5))
28		3orass 1104	. . . . . . 7 ⊢ ((𝑛 = 3 ∨ 𝑛 = 1 ∨ 𝑛 = 5) ↔ (𝑛 = 3 ∨ (𝑛 = 1 ∨ 𝑛 = 5)))
29	27, 28	bitr3i 280	. . . . . 6 ⊢ ((𝑛 = 1 ∨ 𝑛 = 3 ∨ 𝑛 = 5) ↔ (𝑛 = 3 ∨ (𝑛 = 1 ∨ 𝑛 = 5)))
30		vex 3457	. . . . . . 7 ⊢ 𝑛 ∈ V
31	30	eltp 4654	. . . . . 6 ⊢ (𝑛 ∈ {1, 3, 5} ↔ (𝑛 = 1 ∨ 𝑛 = 3 ∨ 𝑛 = 5))
32		prex 5409	. . . . . . . 8 ⊢ {0, 𝑛} ∈ V
33		el7g 4655	. . . . . . . 8 ⊢ ({0, 𝑛} ∈ V → ({0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ ({0, 𝑛} = {0, 3} ∨ (({0, 𝑛} = {0, 1} ∨ {0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3}) ∨ ({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5})))))
34	32, 33	ax-mp 5	. . . . . . 7 ⊢ ({0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ ({0, 𝑛} = {0, 3} ∨ (({0, 𝑛} = {0, 1} ∨ {0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3}) ∨ ({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5}))))
35	30	a1i 11	. . . . . . . . . 10 ⊢ (3 ∈ V → 𝑛 ∈ V)
36		elex 3474	. . . . . . . . . 10 ⊢ (3 ∈ V → 3 ∈ V)
37	35, 36	preq2b 4811	. . . . . . . . 9 ⊢ (3 ∈ V → ({0, 𝑛} = {0, 3} ↔ 𝑛 = 3))
38	15, 37	ax-mp 5	. . . . . . . 8 ⊢ ({0, 𝑛} = {0, 3} ↔ 𝑛 = 3)
39		3orrot 1106	. . . . . . . . . 10 ⊢ (({0, 𝑛} = {0, 1} ∨ {0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3}) ↔ ({0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3} ∨ {0, 𝑛} = {0, 1}))
40	1, 30	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ V ∧ 𝑛 ∈ V)
41		1ex 11202	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ V
42		2ex 12317	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ V
43	41, 42	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V ∧ 2 ∈ V)
44	40, 43	pm3.2i 475	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ V ∧ 𝑛 ∈ V) ∧ (1 ∈ V ∧ 2 ∈ V))
45		0ne1 12311	. . . . . . . . . . . . . . . 16 ⊢ 0 ≠ 1
46		0ne2 12449	. . . . . . . . . . . . . . . 16 ⊢ 0 ≠ 2
47	45, 46	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ (0 ≠ 1 ∧ 0 ≠ 2)
48	47	orci 878	. . . . . . . . . . . . . 14 ⊢ ((0 ≠ 1 ∧ 0 ≠ 2) ∨ (𝑛 ≠ 1 ∧ 𝑛 ≠ 2))
49		prneimg 4818	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ V ∧ 𝑛 ∈ V) ∧ (1 ∈ V ∧ 2 ∈ V)) → (((0 ≠ 1 ∧ 0 ≠ 2) ∨ (𝑛 ≠ 1 ∧ 𝑛 ≠ 2)) → {0, 𝑛} ≠ {1, 2}))
50	44, 48, 49	mp2 9	. . . . . . . . . . . . 13 ⊢ {0, 𝑛} ≠ {1, 2}
51	50	neii 2958	. . . . . . . . . . . 12 ⊢ ¬ {0, 𝑛} = {1, 2}
52		id 23	. . . . . . . . . . . . 13 ⊢ (¬ {0, 𝑛} = {1, 2} → ¬ {0, 𝑛} = {1, 2})
53	42, 15	pm3.2i 475	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ V ∧ 3 ∈ V)
54	40, 53	pm3.2i 475	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ V ∧ 𝑛 ∈ V) ∧ (2 ∈ V ∧ 3 ∈ V))
55		0re 11209	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
56		3pos 12348	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 3
57	55, 56	ltneii 11322	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ≠ 3
58	46, 57	pm3.2i 475	. . . . . . . . . . . . . . . . 17 ⊢ (0 ≠ 2 ∧ 0 ≠ 3)
59	58	orci 878	. . . . . . . . . . . . . . . 16 ⊢ ((0 ≠ 2 ∧ 0 ≠ 3) ∨ (𝑛 ≠ 2 ∧ 𝑛 ≠ 3))
60		prneimg 4818	. . . . . . . . . . . . . . . 16 ⊢ (((0 ∈ V ∧ 𝑛 ∈ V) ∧ (2 ∈ V ∧ 3 ∈ V)) → (((0 ≠ 2 ∧ 0 ≠ 3) ∨ (𝑛 ≠ 2 ∧ 𝑛 ≠ 3)) → {0, 𝑛} ≠ {2, 3}))
61	54, 59, 60	mp2 9	. . . . . . . . . . . . . . 15 ⊢ {0, 𝑛} ≠ {2, 3}
62	61	neii 2958	. . . . . . . . . . . . . 14 ⊢ ¬ {0, 𝑛} = {2, 3}
63	62	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ {0, 𝑛} = {1, 2} → ¬ {0, 𝑛} = {2, 3})
64	52, 63	3bior2fd 1506	. . . . . . . . . . . 12 ⊢ (¬ {0, 𝑛} = {1, 2} → ({0, 𝑛} = {0, 1} ↔ ({0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3} ∨ {0, 𝑛} = {0, 1})))
65	51, 64	ax-mp 5	. . . . . . . . . . 11 ⊢ ({0, 𝑛} = {0, 1} ↔ ({0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3} ∨ {0, 𝑛} = {0, 1}))
66	30	a1i 11	. . . . . . . . . . . . 13 ⊢ (1 ∈ V → 𝑛 ∈ V)
67		elex 3474	. . . . . . . . . . . . 13 ⊢ (1 ∈ V → 1 ∈ V)
68	66, 67	preq2b 4811	. . . . . . . . . . . 12 ⊢ (1 ∈ V → ({0, 𝑛} = {0, 1} ↔ 𝑛 = 1))
69	41, 68	ax-mp 5	. . . . . . . . . . 11 ⊢ ({0, 𝑛} = {0, 1} ↔ 𝑛 = 1)
70	65, 69	bitr3i 280	. . . . . . . . . 10 ⊢ (({0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3} ∨ {0, 𝑛} = {0, 1}) ↔ 𝑛 = 1)
71	39, 70	bitri 278	. . . . . . . . 9 ⊢ (({0, 𝑛} = {0, 1} ∨ {0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3}) ↔ 𝑛 = 1)
72		4nn0 12522	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℕ₀
73	15, 72	pm3.2i 475	. . . . . . . . . . . . . 14 ⊢ (3 ∈ V ∧ 4 ∈ ℕ₀)
74	40, 73	pm3.2i 475	. . . . . . . . . . . . 13 ⊢ ((0 ∈ V ∧ 𝑛 ∈ V) ∧ (3 ∈ V ∧ 4 ∈ ℕ₀))
75		4pos 12350	. . . . . . . . . . . . . . . 16 ⊢ 0 < 4
76	55, 75	ltneii 11322	. . . . . . . . . . . . . . 15 ⊢ 0 ≠ 4
77	57, 76	pm3.2i 475	. . . . . . . . . . . . . 14 ⊢ (0 ≠ 3 ∧ 0 ≠ 4)
78	77	orci 878	. . . . . . . . . . . . 13 ⊢ ((0 ≠ 3 ∧ 0 ≠ 4) ∨ (𝑛 ≠ 3 ∧ 𝑛 ≠ 4))
79		prneimg 4818	. . . . . . . . . . . . 13 ⊢ (((0 ∈ V ∧ 𝑛 ∈ V) ∧ (3 ∈ V ∧ 4 ∈ ℕ₀)) → (((0 ≠ 3 ∧ 0 ≠ 4) ∨ (𝑛 ≠ 3 ∧ 𝑛 ≠ 4)) → {0, 𝑛} ≠ {3, 4}))
80	74, 78, 79	mp2 9	. . . . . . . . . . . 12 ⊢ {0, 𝑛} ≠ {3, 4}
81	80	neii 2958	. . . . . . . . . . 11 ⊢ ¬ {0, 𝑛} = {3, 4}
82		id 23	. . . . . . . . . . . 12 ⊢ (¬ {0, 𝑛} = {3, 4} → ¬ {0, 𝑛} = {3, 4})
83	72, 20	pm3.2i 475	. . . . . . . . . . . . . . . 16 ⊢ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀)
84	40, 83	pm3.2i 475	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ V ∧ 𝑛 ∈ V) ∧ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀))
85		5pos 12352	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 5
86	55, 85	ltneii 11322	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≠ 5
87	76, 86	pm3.2i 475	. . . . . . . . . . . . . . . 16 ⊢ (0 ≠ 4 ∧ 0 ≠ 5)
88	87	orci 878	. . . . . . . . . . . . . . 15 ⊢ ((0 ≠ 4 ∧ 0 ≠ 5) ∨ (𝑛 ≠ 4 ∧ 𝑛 ≠ 5))
89		prneimg 4818	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ V ∧ 𝑛 ∈ V) ∧ (4 ∈ ℕ₀ ∧ 5 ∈ ℕ₀)) → (((0 ≠ 4 ∧ 0 ≠ 5) ∨ (𝑛 ≠ 4 ∧ 𝑛 ≠ 5)) → {0, 𝑛} ≠ {4, 5}))
90	84, 88, 89	mp2 9	. . . . . . . . . . . . . 14 ⊢ {0, 𝑛} ≠ {4, 5}
91	90	neii 2958	. . . . . . . . . . . . 13 ⊢ ¬ {0, 𝑛} = {4, 5}
92	91	a1i 11	. . . . . . . . . . . 12 ⊢ (¬ {0, 𝑛} = {3, 4} → ¬ {0, 𝑛} = {4, 5})
93	82, 92	3bior2fd 1506	. . . . . . . . . . 11 ⊢ (¬ {0, 𝑛} = {3, 4} → ({0, 𝑛} = {0, 5} ↔ ({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5})))
94	81, 93	ax-mp 5	. . . . . . . . . 10 ⊢ ({0, 𝑛} = {0, 5} ↔ ({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5}))
95	30	a1i 11	. . . . . . . . . . . 12 ⊢ (5 ∈ ℕ₀ → 𝑛 ∈ V)
96		elex 3474	. . . . . . . . . . . 12 ⊢ (5 ∈ ℕ₀ → 5 ∈ V)
97	95, 96	preq2b 4811	. . . . . . . . . . 11 ⊢ (5 ∈ ℕ₀ → ({0, 𝑛} = {0, 5} ↔ 𝑛 = 5))
98	20, 97	ax-mp 5	. . . . . . . . . 10 ⊢ ({0, 𝑛} = {0, 5} ↔ 𝑛 = 5)
99	94, 98	bitr3i 280	. . . . . . . . 9 ⊢ (({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5}) ↔ 𝑛 = 5)
100	71, 99	orbi12i 927	. . . . . . . 8 ⊢ ((({0, 𝑛} = {0, 1} ∨ {0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3}) ∨ ({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5})) ↔ (𝑛 = 1 ∨ 𝑛 = 5))
101	38, 100	orbi12i 927	. . . . . . 7 ⊢ (({0, 𝑛} = {0, 3} ∨ (({0, 𝑛} = {0, 1} ∨ {0, 𝑛} = {1, 2} ∨ {0, 𝑛} = {2, 3}) ∨ ({0, 𝑛} = {3, 4} ∨ {0, 𝑛} = {4, 5} ∨ {0, 𝑛} = {0, 5}))) ↔ (𝑛 = 3 ∨ (𝑛 = 1 ∨ 𝑛 = 5)))
102	34, 101	bitri 278	. . . . . 6 ⊢ ({0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})) ↔ (𝑛 = 3 ∨ (𝑛 = 1 ∨ 𝑛 = 5)))
103	29, 31, 102	3bitr4i 306	. . . . 5 ⊢ (𝑛 ∈ {1, 3, 5} ↔ {0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}})))
104	103	a1i 11	. . . 4 ⊢ (((1 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 5 ∈ ({0, 1, 2} ∪ {3, 4, 5})) ∧ 𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5})) → (𝑛 ∈ {1, 3, 5} ↔ {0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))))
105	26, 104	eqrrabd 4039	. . 3 ⊢ ((1 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 3 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∧ 5 ∈ ({0, 1, 2} ∪ {3, 4, 5})) → {1, 3, 5} = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))})
106	14, 19, 25, 105	mp3an 1488	. 2 ⊢ {1, 3, 5} = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {0, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))}
107	10, 106	eqtr4i 2787	1 ⊢ (𝐺 NeighbVtx 0) = {1, 3, 5}

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 {crab 3414 Vcvv 3453 ∪ cun 3902 {csn 4588 {cpr 4590 {ctp 4592 ⟨cop 4594 (class class class)co 7410 0cc0 11099 1c1 11100 2c2 12294 3c3 12295 4c4 12296 5c5 12297 ℕ₀cn0 12503 ...cfz 13534 ⟨“cs7 14883 NeighbVtx cnbgr 29648

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-n0 12504 df-xnn0 12577 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13683 df-hash 14367 df-word 14551 df-concat 14608 df-s1 14634 df-s2 14885 df-s3 14886 df-s4 14887 df-s5 14888 df-s6 14889 df-s7 14890 df-vtx 29314 df-iedg 29315 df-edg 29364 df-upgr 29398 df-umgr 29399 df-usgr 29467 df-nbgr 29649

This theorem is referenced by: usgrexmpl2trifr 48769

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