Metamath Proof Explorer

Theorem usgrexmpl2nb1

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

Ref Expression
Hypotheses 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 usgrexmpl2nb1 ⊢ G NeighbVtx 1 = 0 2

Proof

Step Hyp Ref Expression
1 usgrexmpl2.v ⊢ V = 0 … 5
2 usgrexmpl2.e ⊢ E = ⟨“ 0 1 1 2 2 3 3 4 4 5 0 3 0 5 ”⟩
3 usgrexmpl2.g ⊢ G = V E
4 1eltp012 ⊢ 1 ∈ 0 1 2
5 4 orci ⊢ 1 ∈ 0 1 2 ∨ 1 ∈ 3 4 5
6 elun ⊢ 1 ∈ 0 1 2 ∪ 3 4 5 ↔ 1 ∈ 0 1 2 ∨ 1 ∈ 3 4 5
7 5 6 mpbir ⊢ 1 ∈ 0 1 2 ∪ 3 4 5
8 1 2 3 usgrexmpl2nblem ⊢ 1 ∈ 0 1 2 ∪ 3 4 5 → G NeighbVtx 1 = n ∈ 0 1 2 ∪ 3 4 5 | 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5
9 7 8 ax-mp ⊢ G NeighbVtx 1 = n ∈ 0 1 2 ∪ 3 4 5 | 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5
10 c0ex ⊢ 0 ∈ V
11 10 tpid1 ⊢ 0 ∈ 0 1 2
12 11 orci ⊢ 0 ∈ 0 1 2 ∨ 0 ∈ 3 4 5
13 elun ⊢ 0 ∈ 0 1 2 ∪ 3 4 5 ↔ 0 ∈ 0 1 2 ∨ 0 ∈ 3 4 5
14 12 13 mpbir ⊢ 0 ∈ 0 1 2 ∪ 3 4 5
15 2ex ⊢ 2 ∈ V
16 15 tpid3 ⊢ 2 ∈ 0 1 2
17 16 orci ⊢ 2 ∈ 0 1 2 ∨ 2 ∈ 3 4 5
18 elun ⊢ 2 ∈ 0 1 2 ∪ 3 4 5 ↔ 2 ∈ 0 1 2 ∨ 2 ∈ 3 4 5
19 17 18 mpbir ⊢ 2 ∈ 0 1 2 ∪ 3 4 5
20 prssi ⊢ 0 ∈ 0 1 2 ∪ 3 4 5 ∧ 2 ∈ 0 1 2 ∪ 3 4 5 → 0 2 ⊆ 0 1 2 ∪ 3 4 5
21 1re ⊢ 1 ∈ ℝ
22 vex ⊢ n ∈ V
23 21 22 pm3.2i ⊢ 1 ∈ ℝ ∧ n ∈ V
24 3ex ⊢ 3 ∈ V
25 15 24 pm3.2i ⊢ 2 ∈ V ∧ 3 ∈ V
26 23 25 pm3.2i ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 2 ∈ V ∧ 3 ∈ V
27 1ne2 ⊢ 1 ≠ 2
28 1lt3 ⊢ 1 < 3
29 21 28 ltneii ⊢ 1 ≠ 3
30 27 29 pm3.2i ⊢ 1 ≠ 2 ∧ 1 ≠ 3
31 30 orci ⊢ 1 ≠ 2 ∧ 1 ≠ 3 ∨ n ≠ 2 ∧ n ≠ 3
32 prneimg ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 2 ∈ V ∧ 3 ∈ V → 1 ≠ 2 ∧ 1 ≠ 3 ∨ n ≠ 2 ∧ n ≠ 3 → 1 n ≠ 2 3
33 26 31 32 mp2 ⊢ 1 n ≠ 2 3
34 33 neii ⊢ ¬ 1 n = 2 3
35 34 biorfri ⊢ 1 n = 0 1 ∨ 1 n = 1 2 ↔ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3
36 22 a1i ⊢ 0 ∈ V → n ∈ V
37 elex ⊢ 0 ∈ V → 0 ∈ V
38 36 37 preq2b ⊢ 0 ∈ V → 1 n = 1 0 ↔ n = 0
39 10 38 ax-mp ⊢ 1 n = 1 0 ↔ n = 0
40 prcom ⊢ 1 0 = 0 1
41 40 eqeq2i ⊢ 1 n = 1 0 ↔ 1 n = 0 1
42 39 41 bitr3i ⊢ n = 0 ↔ 1 n = 0 1
43 22 a1i ⊢ 2 ∈ V → n ∈ V
44 elex ⊢ 2 ∈ V → 2 ∈ V
45 43 44 preq2b ⊢ 2 ∈ V → 1 n = 1 2 ↔ n = 2
46 45 bicomd ⊢ 2 ∈ V → n = 2 ↔ 1 n = 1 2
47 15 46 ax-mp ⊢ n = 2 ↔ 1 n = 1 2
48 42 47 orbi12i ⊢ n = 0 ∨ n = 2 ↔ 1 n = 0 1 ∨ 1 n = 1 2
49 df-3or ⊢ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ↔ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3
50 35 48 49 3bitr4i ⊢ n = 0 ∨ n = 2 ↔ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3
51 4nn0 ⊢ 4 ∈ ℕ 0
52 24 51 pm3.2i ⊢ 3 ∈ V ∧ 4 ∈ ℕ 0
53 23 52 pm3.2i ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 3 ∈ V ∧ 4 ∈ ℕ 0
54 1lt4 ⊢ 1 < 4
55 21 54 ltneii ⊢ 1 ≠ 4
56 29 55 pm3.2i ⊢ 1 ≠ 3 ∧ 1 ≠ 4
57 56 orci ⊢ 1 ≠ 3 ∧ 1 ≠ 4 ∨ n ≠ 3 ∧ n ≠ 4
58 prneimg ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 3 ∈ V ∧ 4 ∈ ℕ 0 → 1 ≠ 3 ∧ 1 ≠ 4 ∨ n ≠ 3 ∧ n ≠ 4 → 1 n ≠ 3 4
59 53 57 58 mp2 ⊢ 1 n ≠ 3 4
60 59 neii ⊢ ¬ 1 n = 3 4
61 5nn0 ⊢ 5 ∈ ℕ 0
62 51 61 pm3.2i ⊢ 4 ∈ ℕ 0 ∧ 5 ∈ ℕ 0
63 23 62 pm3.2i ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 4 ∈ ℕ 0 ∧ 5 ∈ ℕ 0
64 1lt5 ⊢ 1 < 5
65 21 64 ltneii ⊢ 1 ≠ 5
66 55 65 pm3.2i ⊢ 1 ≠ 4 ∧ 1 ≠ 5
67 66 orci ⊢ 1 ≠ 4 ∧ 1 ≠ 5 ∨ n ≠ 4 ∧ n ≠ 5
68 prneimg ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 4 ∈ ℕ 0 ∧ 5 ∈ ℕ 0 → 1 ≠ 4 ∧ 1 ≠ 5 ∨ n ≠ 4 ∧ n ≠ 5 → 1 n ≠ 4 5
69 63 67 68 mp2 ⊢ 1 n ≠ 4 5
70 69 neii ⊢ ¬ 1 n = 4 5
71 10 61 pm3.2i ⊢ 0 ∈ V ∧ 5 ∈ ℕ 0
72 23 71 pm3.2i ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 0 ∈ V ∧ 5 ∈ ℕ 0
73 ax-1ne0 ⊢ 1 ≠ 0
74 73 65 pm3.2i ⊢ 1 ≠ 0 ∧ 1 ≠ 5
75 74 orci ⊢ 1 ≠ 0 ∧ 1 ≠ 5 ∨ n ≠ 0 ∧ n ≠ 5
76 prneimg ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 0 ∈ V ∧ 5 ∈ ℕ 0 → 1 ≠ 0 ∧ 1 ≠ 5 ∨ n ≠ 0 ∧ n ≠ 5 → 1 n ≠ 0 5
77 72 75 76 mp2 ⊢ 1 n ≠ 0 5
78 77 neii ⊢ ¬ 1 n = 0 5
79 60 70 78 3pm3.2ni ⊢ ¬ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5
80 79 biorfri ⊢ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ↔ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ∨ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5
81 10 24 pm3.2i ⊢ 0 ∈ V ∧ 3 ∈ V
82 23 81 pm3.2i ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 0 ∈ V ∧ 3 ∈ V
83 73 29 pm3.2i ⊢ 1 ≠ 0 ∧ 1 ≠ 3
84 83 orci ⊢ 1 ≠ 0 ∧ 1 ≠ 3 ∨ n ≠ 0 ∧ n ≠ 3
85 prneimg ⊢ 1 ∈ ℝ ∧ n ∈ V ∧ 0 ∈ V ∧ 3 ∈ V → 1 ≠ 0 ∧ 1 ≠ 3 ∨ n ≠ 0 ∧ n ≠ 3 → 1 n ≠ 0 3
86 82 84 85 mp2 ⊢ 1 n ≠ 0 3
87 86 neii ⊢ ¬ 1 n = 0 3
88 87 biorfi ⊢ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ∨ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5 ↔ 1 n = 0 3 ∨ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ∨ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5
89 50 80 88 3bitri ⊢ n = 0 ∨ n = 2 ↔ 1 n = 0 3 ∨ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ∨ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5
90 22 elpr ⊢ n ∈ 0 2 ↔ n = 0 ∨ n = 2
91 prex ⊢ 1 n ∈ V
92 el7g ⊢ 1 n ∈ V → 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5 ↔ 1 n = 0 3 ∨ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ∨ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5
93 91 92 ax-mp ⊢ 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5 ↔ 1 n = 0 3 ∨ 1 n = 0 1 ∨ 1 n = 1 2 ∨ 1 n = 2 3 ∨ 1 n = 3 4 ∨ 1 n = 4 5 ∨ 1 n = 0 5
94 89 90 93 3bitr4i ⊢ n ∈ 0 2 ↔ 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5
95 94 a1i ⊢ 0 ∈ 0 1 2 ∪ 3 4 5 ∧ 2 ∈ 0 1 2 ∪ 3 4 5 ∧ n ∈ 0 1 2 ∪ 3 4 5 → n ∈ 0 2 ↔ 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5
96 20 95 eqrrabd ⊢ 0 ∈ 0 1 2 ∪ 3 4 5 ∧ 2 ∈ 0 1 2 ∪ 3 4 5 → 0 2 = n ∈ 0 1 2 ∪ 3 4 5 | 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5
97 96 eqcomd ⊢ 0 ∈ 0 1 2 ∪ 3 4 5 ∧ 2 ∈ 0 1 2 ∪ 3 4 5 → n ∈ 0 1 2 ∪ 3 4 5 | 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5 = 0 2
98 14 19 97 mp2an ⊢ n ∈ 0 1 2 ∪ 3 4 5 | 1 n ∈ 0 3 ∪ 0 1 1 2 2 3 ∪ 3 4 4 5 0 5 = 0 2
99 9 98 eqtri ⊢ G NeighbVtx 1 = 0 2