Proof of Theorem usgrexmpl2lem
Step | Hyp | Ref
| Expression |
1 | | prex 5455 |
. . . . 5
⊢ {0, 1}
∈ V |
2 | | prex 5455 |
. . . . 5
⊢ {1, 2}
∈ V |
3 | | prex 5455 |
. . . . 5
⊢ {2, 3}
∈ V |
4 | 1, 2, 3 | 3pm3.2i 1339 |
. . . 4
⊢ ({0, 1}
∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) |
5 | | prex 5455 |
. . . 4
⊢ {3, 4}
∈ V |
6 | | prex 5455 |
. . . . 5
⊢ {4, 5}
∈ V |
7 | | prex 5455 |
. . . . 5
⊢ {0, 3}
∈ V |
8 | | prex 5455 |
. . . . 5
⊢ {0, 5}
∈ V |
9 | 6, 7, 8 | 3pm3.2i 1339 |
. . . 4
⊢ ({4, 5}
∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V) |
10 | 4, 5, 9 | 3pm3.2i 1339 |
. . 3
⊢ (({0, 1}
∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈
V)) |
11 | | 0nn0 12564 |
. . . . . . . . . 10
⊢ 0 ∈
ℕ0 |
12 | | 1nn0 12565 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ0 |
13 | 11, 12 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ∈
ℕ0 ∧ 1 ∈ ℕ0) |
14 | | 2nn0 12566 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
15 | 12, 14 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ∈
ℕ0 ∧ 2 ∈ ℕ0) |
16 | 13, 15 | pm3.2i 470 |
. . . . . . . 8
⊢ ((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (1 ∈
ℕ0 ∧ 2 ∈ ℕ0)) |
17 | | 0ne1 12360 |
. . . . . . . . . 10
⊢ 0 ≠
1 |
18 | | 0ne2 12496 |
. . . . . . . . . 10
⊢ 0 ≠
2 |
19 | 17, 18 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ≠ 1
∧ 0 ≠ 2) |
20 | 19 | orci 864 |
. . . . . . . 8
⊢ ((0 ≠
1 ∧ 0 ≠ 2) ∨ (1 ≠ 1 ∧ 1 ≠ 2)) |
21 | | prneimg 4879 |
. . . . . . . 8
⊢ (((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (1 ∈
ℕ0 ∧ 2 ∈ ℕ0)) → (((0 ≠ 1
∧ 0 ≠ 2) ∨ (1 ≠ 1 ∧ 1 ≠ 2)) → {0, 1} ≠ {1,
2})) |
22 | 16, 20, 21 | mp2 9 |
. . . . . . 7
⊢ {0, 1}
≠ {1, 2} |
23 | | 3nn0 12567 |
. . . . . . . . . 10
⊢ 3 ∈
ℕ0 |
24 | 14, 23 | pm3.2i 470 |
. . . . . . . . 9
⊢ (2 ∈
ℕ0 ∧ 3 ∈ ℕ0) |
25 | 13, 24 | pm3.2i 470 |
. . . . . . . 8
⊢ ((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (2 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
26 | | 0re 11288 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
27 | | 3pos 12394 |
. . . . . . . . . . 11
⊢ 0 <
3 |
28 | 26, 27 | ltneii 11399 |
. . . . . . . . . 10
⊢ 0 ≠
3 |
29 | 18, 28 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ≠ 2
∧ 0 ≠ 3) |
30 | 29 | orci 864 |
. . . . . . . 8
⊢ ((0 ≠
2 ∧ 0 ≠ 3) ∨ (1 ≠ 2 ∧ 1 ≠ 3)) |
31 | | prneimg 4879 |
. . . . . . . 8
⊢ (((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (2 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((0 ≠ 2
∧ 0 ≠ 3) ∨ (1 ≠ 2 ∧ 1 ≠ 3)) → {0, 1} ≠ {2,
3})) |
32 | 25, 30, 31 | mp2 9 |
. . . . . . 7
⊢ {0, 1}
≠ {2, 3} |
33 | | 4nn0 12568 |
. . . . . . . . . 10
⊢ 4 ∈
ℕ0 |
34 | 23, 33 | pm3.2i 470 |
. . . . . . . . 9
⊢ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0) |
35 | 13, 34 | pm3.2i 470 |
. . . . . . . 8
⊢ ((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0)) |
36 | | 4pos 12396 |
. . . . . . . . . . 11
⊢ 0 <
4 |
37 | 26, 36 | ltneii 11399 |
. . . . . . . . . 10
⊢ 0 ≠
4 |
38 | 28, 37 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ≠ 3
∧ 0 ≠ 4) |
39 | 38 | orci 864 |
. . . . . . . 8
⊢ ((0 ≠
3 ∧ 0 ≠ 4) ∨ (1 ≠ 3 ∧ 1 ≠ 4)) |
40 | | prneimg 4879 |
. . . . . . . 8
⊢ (((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0)) → (((0 ≠ 3
∧ 0 ≠ 4) ∨ (1 ≠ 3 ∧ 1 ≠ 4)) → {0, 1} ≠ {3,
4})) |
41 | 35, 39, 40 | mp2 9 |
. . . . . . 7
⊢ {0, 1}
≠ {3, 4} |
42 | 22, 32, 41 | 3pm3.2i 1339 |
. . . . . 6
⊢ ({0, 1}
≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) |
43 | | 5nn0 12569 |
. . . . . . . . . 10
⊢ 5 ∈
ℕ0 |
44 | 33, 43 | pm3.2i 470 |
. . . . . . . . 9
⊢ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0) |
45 | 13, 44 | pm3.2i 470 |
. . . . . . . 8
⊢ ((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
46 | | 5pos 12398 |
. . . . . . . . . . 11
⊢ 0 <
5 |
47 | 26, 46 | ltneii 11399 |
. . . . . . . . . 10
⊢ 0 ≠
5 |
48 | 37, 47 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ≠ 4
∧ 0 ≠ 5) |
49 | 48 | orci 864 |
. . . . . . . 8
⊢ ((0 ≠
4 ∧ 0 ≠ 5) ∨ (1 ≠ 4 ∧ 1 ≠ 5)) |
50 | | prneimg 4879 |
. . . . . . . 8
⊢ (((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((0 ≠ 4
∧ 0 ≠ 5) ∨ (1 ≠ 4 ∧ 1 ≠ 5)) → {0, 1} ≠ {4,
5})) |
51 | 45, 49, 50 | mp2 9 |
. . . . . . 7
⊢ {0, 1}
≠ {4, 5} |
52 | 11, 23 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0) |
53 | 13, 52 | pm3.2i 470 |
. . . . . . . 8
⊢ ((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
54 | | ax-1ne0 11249 |
. . . . . . . . . 10
⊢ 1 ≠
0 |
55 | | 1re 11286 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ |
56 | | 1lt3 12462 |
. . . . . . . . . . 11
⊢ 1 <
3 |
57 | 55, 56 | ltneii 11399 |
. . . . . . . . . 10
⊢ 1 ≠
3 |
58 | 54, 57 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ≠ 0
∧ 1 ≠ 3) |
59 | 58 | olci 865 |
. . . . . . . 8
⊢ ((0 ≠
0 ∧ 0 ≠ 3) ∨ (1 ≠ 0 ∧ 1 ≠ 3)) |
60 | | prneimg 4879 |
. . . . . . . 8
⊢ (((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((0 ≠ 0
∧ 0 ≠ 3) ∨ (1 ≠ 0 ∧ 1 ≠ 3)) → {0, 1} ≠ {0,
3})) |
61 | 53, 59, 60 | mp2 9 |
. . . . . . 7
⊢ {0, 1}
≠ {0, 3} |
62 | 11, 43 | pm3.2i 470 |
. . . . . . . . 9
⊢ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0) |
63 | 13, 62 | pm3.2i 470 |
. . . . . . . 8
⊢ ((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
64 | | 1lt5 12469 |
. . . . . . . . . . 11
⊢ 1 <
5 |
65 | 55, 64 | ltneii 11399 |
. . . . . . . . . 10
⊢ 1 ≠
5 |
66 | 54, 65 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ≠ 0
∧ 1 ≠ 5) |
67 | 66 | olci 865 |
. . . . . . . 8
⊢ ((0 ≠
0 ∧ 0 ≠ 5) ∨ (1 ≠ 0 ∧ 1 ≠ 5)) |
68 | | prneimg 4879 |
. . . . . . . 8
⊢ (((0
∈ ℕ0 ∧ 1 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((0 ≠ 0
∧ 0 ≠ 5) ∨ (1 ≠ 0 ∧ 1 ≠ 5)) → {0, 1} ≠ {0,
5})) |
69 | 63, 67, 68 | mp2 9 |
. . . . . . 7
⊢ {0, 1}
≠ {0, 5} |
70 | 51, 61, 69 | 3pm3.2i 1339 |
. . . . . 6
⊢ ({0, 1}
≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5}) |
71 | 42, 70 | pm3.2i 470 |
. . . . 5
⊢ (({0, 1}
≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧ ({0, 1}
≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0,
5})) |
72 | 15, 24 | pm3.2i 470 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (2 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
73 | | 1ne2 12497 |
. . . . . . . . . 10
⊢ 1 ≠
2 |
74 | 73, 57 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ≠ 2
∧ 1 ≠ 3) |
75 | 74 | orci 864 |
. . . . . . . 8
⊢ ((1 ≠
2 ∧ 1 ≠ 3) ∨ (2 ≠ 2 ∧ 2 ≠ 3)) |
76 | | prneimg 4879 |
. . . . . . . 8
⊢ (((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (2 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((1 ≠ 2
∧ 1 ≠ 3) ∨ (2 ≠ 2 ∧ 2 ≠ 3)) → {1, 2} ≠ {2,
3})) |
77 | 72, 75, 76 | mp2 9 |
. . . . . . 7
⊢ {1, 2}
≠ {2, 3} |
78 | 15, 34 | pm3.2i 470 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0)) |
79 | | 1lt4 12465 |
. . . . . . . . . . 11
⊢ 1 <
4 |
80 | 55, 79 | ltneii 11399 |
. . . . . . . . . 10
⊢ 1 ≠
4 |
81 | 57, 80 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ≠ 3
∧ 1 ≠ 4) |
82 | 81 | orci 864 |
. . . . . . . 8
⊢ ((1 ≠
3 ∧ 1 ≠ 4) ∨ (2 ≠ 3 ∧ 2 ≠ 4)) |
83 | | prneimg 4879 |
. . . . . . . 8
⊢ (((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0)) → (((1 ≠ 3
∧ 1 ≠ 4) ∨ (2 ≠ 3 ∧ 2 ≠ 4)) → {1, 2} ≠ {3,
4})) |
84 | 78, 82, 83 | mp2 9 |
. . . . . . 7
⊢ {1, 2}
≠ {3, 4} |
85 | 77, 84 | pm3.2i 470 |
. . . . . 6
⊢ ({1, 2}
≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) |
86 | 15, 44 | pm3.2i 470 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
87 | 80, 65 | pm3.2i 470 |
. . . . . . . . 9
⊢ (1 ≠ 4
∧ 1 ≠ 5) |
88 | 87 | orci 864 |
. . . . . . . 8
⊢ ((1 ≠
4 ∧ 1 ≠ 5) ∨ (2 ≠ 4 ∧ 2 ≠ 5)) |
89 | | prneimg 4879 |
. . . . . . . 8
⊢ (((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((1 ≠ 4
∧ 1 ≠ 5) ∨ (2 ≠ 4 ∧ 2 ≠ 5)) → {1, 2} ≠ {4,
5})) |
90 | 86, 88, 89 | mp2 9 |
. . . . . . 7
⊢ {1, 2}
≠ {4, 5} |
91 | 15, 52 | pm3.2i 470 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
92 | 58 | orci 864 |
. . . . . . . 8
⊢ ((1 ≠
0 ∧ 1 ≠ 3) ∨ (2 ≠ 0 ∧ 2 ≠ 3)) |
93 | | prneimg 4879 |
. . . . . . . 8
⊢ (((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((1 ≠ 0
∧ 1 ≠ 3) ∨ (2 ≠ 0 ∧ 2 ≠ 3)) → {1, 2} ≠ {0,
3})) |
94 | 91, 92, 93 | mp2 9 |
. . . . . . 7
⊢ {1, 2}
≠ {0, 3} |
95 | 15, 62 | pm3.2i 470 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
96 | 66 | orci 864 |
. . . . . . . 8
⊢ ((1 ≠
0 ∧ 1 ≠ 5) ∨ (2 ≠ 0 ∧ 2 ≠ 5)) |
97 | | prneimg 4879 |
. . . . . . . 8
⊢ (((1
∈ ℕ0 ∧ 2 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((1 ≠ 0
∧ 1 ≠ 5) ∨ (2 ≠ 0 ∧ 2 ≠ 5)) → {1, 2} ≠ {0,
5})) |
98 | 95, 96, 97 | mp2 9 |
. . . . . . 7
⊢ {1, 2}
≠ {0, 5} |
99 | 90, 94, 98 | 3pm3.2i 1339 |
. . . . . 6
⊢ ({1, 2}
≠ {4, 5} ∧ {1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5}) |
100 | 85, 99 | pm3.2i 470 |
. . . . 5
⊢ (({1, 2}
≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧ {1, 2}
≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) |
101 | 24, 34 | pm3.2i 470 |
. . . . . . 7
⊢ ((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0)) |
102 | | 2re 12363 |
. . . . . . . . . 10
⊢ 2 ∈
ℝ |
103 | | 2lt3 12461 |
. . . . . . . . . 10
⊢ 2 <
3 |
104 | 102, 103 | ltneii 11399 |
. . . . . . . . 9
⊢ 2 ≠
3 |
105 | | 2lt4 12464 |
. . . . . . . . . 10
⊢ 2 <
4 |
106 | 102, 105 | ltneii 11399 |
. . . . . . . . 9
⊢ 2 ≠
4 |
107 | 104, 106 | pm3.2i 470 |
. . . . . . . 8
⊢ (2 ≠ 3
∧ 2 ≠ 4) |
108 | 107 | orci 864 |
. . . . . . 7
⊢ ((2 ≠
3 ∧ 2 ≠ 4) ∨ (3 ≠ 3 ∧ 3 ≠ 4)) |
109 | | prneimg 4879 |
. . . . . . 7
⊢ (((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (3 ∈
ℕ0 ∧ 4 ∈ ℕ0)) → (((2 ≠ 3
∧ 2 ≠ 4) ∨ (3 ≠ 3 ∧ 3 ≠ 4)) → {2, 3} ≠ {3,
4})) |
110 | 101, 108,
109 | mp2 9 |
. . . . . 6
⊢ {2, 3}
≠ {3, 4} |
111 | 24, 44 | pm3.2i 470 |
. . . . . . . 8
⊢ ((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
112 | | 2lt5 12468 |
. . . . . . . . . . 11
⊢ 2 <
5 |
113 | 102, 112 | ltneii 11399 |
. . . . . . . . . 10
⊢ 2 ≠
5 |
114 | 106, 113 | pm3.2i 470 |
. . . . . . . . 9
⊢ (2 ≠ 4
∧ 2 ≠ 5) |
115 | 114 | orci 864 |
. . . . . . . 8
⊢ ((2 ≠
4 ∧ 2 ≠ 5) ∨ (3 ≠ 4 ∧ 3 ≠ 5)) |
116 | | prneimg 4879 |
. . . . . . . 8
⊢ (((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((2 ≠ 4
∧ 2 ≠ 5) ∨ (3 ≠ 4 ∧ 3 ≠ 5)) → {2, 3} ≠ {4,
5})) |
117 | 111, 115,
116 | mp2 9 |
. . . . . . 7
⊢ {2, 3}
≠ {4, 5} |
118 | 24, 52 | pm3.2i 470 |
. . . . . . . 8
⊢ ((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
119 | | 2ne0 12393 |
. . . . . . . . . 10
⊢ 2 ≠
0 |
120 | 119, 104 | pm3.2i 470 |
. . . . . . . . 9
⊢ (2 ≠ 0
∧ 2 ≠ 3) |
121 | 120 | orci 864 |
. . . . . . . 8
⊢ ((2 ≠
0 ∧ 2 ≠ 3) ∨ (3 ≠ 0 ∧ 3 ≠ 3)) |
122 | | prneimg 4879 |
. . . . . . . 8
⊢ (((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((2 ≠ 0
∧ 2 ≠ 3) ∨ (3 ≠ 0 ∧ 3 ≠ 3)) → {2, 3} ≠ {0,
3})) |
123 | 118, 121,
122 | mp2 9 |
. . . . . . 7
⊢ {2, 3}
≠ {0, 3} |
124 | 24, 62 | pm3.2i 470 |
. . . . . . . 8
⊢ ((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
125 | 119, 113 | pm3.2i 470 |
. . . . . . . . 9
⊢ (2 ≠ 0
∧ 2 ≠ 5) |
126 | 125 | orci 864 |
. . . . . . . 8
⊢ ((2 ≠
0 ∧ 2 ≠ 5) ∨ (3 ≠ 0 ∧ 3 ≠ 5)) |
127 | | prneimg 4879 |
. . . . . . . 8
⊢ (((2
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((2 ≠ 0
∧ 2 ≠ 5) ∨ (3 ≠ 0 ∧ 3 ≠ 5)) → {2, 3} ≠ {0,
5})) |
128 | 124, 126,
127 | mp2 9 |
. . . . . . 7
⊢ {2, 3}
≠ {0, 5} |
129 | 117, 123,
128 | 3pm3.2i 1339 |
. . . . . 6
⊢ ({2, 3}
≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}) |
130 | 110, 129 | pm3.2i 470 |
. . . . 5
⊢ ({2, 3}
≠ {3, 4} ∧ ({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3}
≠ {0, 5})) |
131 | 71, 100, 130 | 3pm3.2i 1339 |
. . . 4
⊢ ((({0, 1}
≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧ ({0, 1}
≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧ (({1, 2}
≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧ {1, 2}
≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧ ({2, 3}
≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0,
5}))) |
132 | 34, 44 | pm3.2i 470 |
. . . . . . 7
⊢ ((3
∈ ℕ0 ∧ 4 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
133 | | 3re 12369 |
. . . . . . . . . 10
⊢ 3 ∈
ℝ |
134 | | 3lt4 12463 |
. . . . . . . . . 10
⊢ 3 <
4 |
135 | 133, 134 | ltneii 11399 |
. . . . . . . . 9
⊢ 3 ≠
4 |
136 | | 3lt5 12467 |
. . . . . . . . . 10
⊢ 3 <
5 |
137 | 133, 136 | ltneii 11399 |
. . . . . . . . 9
⊢ 3 ≠
5 |
138 | 135, 137 | pm3.2i 470 |
. . . . . . . 8
⊢ (3 ≠ 4
∧ 3 ≠ 5) |
139 | 138 | orci 864 |
. . . . . . 7
⊢ ((3 ≠
4 ∧ 3 ≠ 5) ∨ (4 ≠ 4 ∧ 4 ≠ 5)) |
140 | | prneimg 4879 |
. . . . . . 7
⊢ (((3
∈ ℕ0 ∧ 4 ∈ ℕ0) ∧ (4 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((3 ≠ 4
∧ 3 ≠ 5) ∨ (4 ≠ 4 ∧ 4 ≠ 5)) → {3, 4} ≠ {4,
5})) |
141 | 132, 139,
140 | mp2 9 |
. . . . . 6
⊢ {3, 4}
≠ {4, 5} |
142 | 34, 52 | pm3.2i 470 |
. . . . . . 7
⊢ ((3
∈ ℕ0 ∧ 4 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
143 | | 4ne0 12397 |
. . . . . . . . 9
⊢ 4 ≠
0 |
144 | 135 | necomi 2997 |
. . . . . . . . 9
⊢ 4 ≠
3 |
145 | 143, 144 | pm3.2i 470 |
. . . . . . . 8
⊢ (4 ≠ 0
∧ 4 ≠ 3) |
146 | 145 | olci 865 |
. . . . . . 7
⊢ ((3 ≠
0 ∧ 3 ≠ 3) ∨ (4 ≠ 0 ∧ 4 ≠ 3)) |
147 | | prneimg 4879 |
. . . . . . 7
⊢ (((3
∈ ℕ0 ∧ 4 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((3 ≠ 0
∧ 3 ≠ 3) ∨ (4 ≠ 0 ∧ 4 ≠ 3)) → {3, 4} ≠ {0,
3})) |
148 | 142, 146,
147 | mp2 9 |
. . . . . 6
⊢ {3, 4}
≠ {0, 3} |
149 | 34, 62 | pm3.2i 470 |
. . . . . . 7
⊢ ((3
∈ ℕ0 ∧ 4 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
150 | | 3ne0 12395 |
. . . . . . . . 9
⊢ 3 ≠
0 |
151 | 150, 137 | pm3.2i 470 |
. . . . . . . 8
⊢ (3 ≠ 0
∧ 3 ≠ 5) |
152 | 151 | orci 864 |
. . . . . . 7
⊢ ((3 ≠
0 ∧ 3 ≠ 5) ∨ (4 ≠ 0 ∧ 4 ≠ 5)) |
153 | | prneimg 4879 |
. . . . . . 7
⊢ (((3
∈ ℕ0 ∧ 4 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((3 ≠ 0
∧ 3 ≠ 5) ∨ (4 ≠ 0 ∧ 4 ≠ 5)) → {3, 4} ≠ {0,
5})) |
154 | 149, 152,
153 | mp2 9 |
. . . . . 6
⊢ {3, 4}
≠ {0, 5} |
155 | 141, 148,
154 | 3pm3.2i 1339 |
. . . . 5
⊢ ({3, 4}
≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) |
156 | 44, 52 | pm3.2i 470 |
. . . . . . 7
⊢ ((4
∈ ℕ0 ∧ 5 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) |
157 | 145 | orci 864 |
. . . . . . 7
⊢ ((4 ≠
0 ∧ 4 ≠ 3) ∨ (5 ≠ 0 ∧ 5 ≠ 3)) |
158 | | prneimg 4879 |
. . . . . . 7
⊢ (((4
∈ ℕ0 ∧ 5 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 3 ∈ ℕ0)) → (((4 ≠ 0
∧ 4 ≠ 3) ∨ (5 ≠ 0 ∧ 5 ≠ 3)) → {4, 5} ≠ {0,
3})) |
159 | 156, 157,
158 | mp2 9 |
. . . . . 6
⊢ {4, 5}
≠ {0, 3} |
160 | 44, 62 | pm3.2i 470 |
. . . . . . 7
⊢ ((4
∈ ℕ0 ∧ 5 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
161 | | 4re 12373 |
. . . . . . . . . 10
⊢ 4 ∈
ℝ |
162 | | 4lt5 12466 |
. . . . . . . . . 10
⊢ 4 <
5 |
163 | 161, 162 | ltneii 11399 |
. . . . . . . . 9
⊢ 4 ≠
5 |
164 | 143, 163 | pm3.2i 470 |
. . . . . . . 8
⊢ (4 ≠ 0
∧ 4 ≠ 5) |
165 | 164 | orci 864 |
. . . . . . 7
⊢ ((4 ≠
0 ∧ 4 ≠ 5) ∨ (5 ≠ 0 ∧ 5 ≠ 5)) |
166 | | prneimg 4879 |
. . . . . . 7
⊢ (((4
∈ ℕ0 ∧ 5 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((4 ≠ 0
∧ 4 ≠ 5) ∨ (5 ≠ 0 ∧ 5 ≠ 5)) → {4, 5} ≠ {0,
5})) |
167 | 160, 165,
166 | mp2 9 |
. . . . . 6
⊢ {4, 5}
≠ {0, 5} |
168 | 52, 62 | pm3.2i 470 |
. . . . . . 7
⊢ ((0
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) |
169 | 151 | olci 865 |
. . . . . . 7
⊢ ((0 ≠
0 ∧ 0 ≠ 5) ∨ (3 ≠ 0 ∧ 3 ≠ 5)) |
170 | | prneimg 4879 |
. . . . . . 7
⊢ (((0
∈ ℕ0 ∧ 3 ∈ ℕ0) ∧ (0 ∈
ℕ0 ∧ 5 ∈ ℕ0)) → (((0 ≠ 0
∧ 0 ≠ 5) ∨ (3 ≠ 0 ∧ 3 ≠ 5)) → {0, 3} ≠ {0,
5})) |
171 | 168, 169,
170 | mp2 9 |
. . . . . 6
⊢ {0, 3}
≠ {0, 5} |
172 | 159, 167,
171 | 3pm3.2i 1339 |
. . . . 5
⊢ ({4, 5}
≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5}) |
173 | 155, 172 | pm3.2i 470 |
. . . 4
⊢ (({3, 4}
≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧ ({4, 5}
≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0,
5})) |
174 | 131, 173 | pm3.2i 470 |
. . 3
⊢ (((({0,
1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧ ({0, 1}
≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧ (({1, 2}
≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧ {1, 2}
≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧ ({2, 3}
≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧ (({3, 4}
≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧ ({4, 5}
≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0,
5}))) |
175 | 10, 174 | pm3.2i 470 |
. 2
⊢ ((({0, 1}
∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0,
5})))) |
176 | | usgrexmpl2.e |
. 2
⊢ 𝐸 = 〈“{0, 1} {1, 2}
{2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 |
177 | | s7f1o 15011 |
. . . . . . 7
⊢ (((({0,
1} ∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5}))))
→ (𝐸 =
〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉
→ 𝐸:(0..^7)–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}))) |
178 | 177 | imp 406 |
. . . . . 6
⊢ ((((({0,
1} ∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5})))) ∧
𝐸 = 〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉) → 𝐸:(0..^7)–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
179 | | s7len 14947 |
. . . . . . . 8
⊢
(♯‘〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3}
{0, 5}”〉) = 7 |
180 | 179 | oveq2i 7456 |
. . . . . . 7
⊢
(0..^(♯‘〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0,
3} {0, 5}”〉)) = (0..^7) |
181 | | f1oeq2 6850 |
. . . . . . 7
⊢
((0..^(♯‘〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0,
3} {0, 5}”〉)) = (0..^7) → (𝐸:(0..^(♯‘〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉))–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}) ↔ 𝐸:(0..^7)–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}))) |
182 | 180, 181 | ax-mp 5 |
. . . . . 6
⊢ (𝐸:(0..^(♯‘〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉))–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}) ↔ 𝐸:(0..^7)–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
183 | 178, 182 | sylibr 234 |
. . . . 5
⊢ ((((({0,
1} ∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5})))) ∧
𝐸 = 〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉) → 𝐸:(0..^(♯‘〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉))–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
184 | 176 | dmeqi 5928 |
. . . . . . 7
⊢ dom 𝐸 = dom 〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 |
185 | | s7cli 14930 |
. . . . . . . 8
⊢
〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0,
5}”〉 ∈ Word V |
186 | | wrddm 14565 |
. . . . . . . 8
⊢
(〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0,
5}”〉 ∈ Word V → dom 〈“{0, 1} {1, 2} {2, 3} {3,
4} {4, 5} {0, 3} {0, 5}”〉 = (0..^(♯‘〈“{0, 1}
{1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉))) |
187 | 185, 186 | ax-mp 5 |
. . . . . . 7
⊢ dom
〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉 =
(0..^(♯‘〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0,
5}”〉)) |
188 | 184, 187 | eqtri 2762 |
. . . . . 6
⊢ dom 𝐸 =
(0..^(♯‘〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0,
5}”〉)) |
189 | | f1oeq2 6850 |
. . . . . 6
⊢ (dom
𝐸 =
(0..^(♯‘〈“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0,
5}”〉)) → (𝐸:dom 𝐸–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}) ↔ 𝐸:(0..^(♯‘〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉))–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}))) |
190 | 188, 189 | ax-mp 5 |
. . . . 5
⊢ (𝐸:dom 𝐸–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}) ↔ 𝐸:(0..^(♯‘〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉))–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
191 | 183, 190 | sylibr 234 |
. . . 4
⊢ ((((({0,
1} ∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5})))) ∧
𝐸 = 〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉) → 𝐸:dom 𝐸–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
192 | | f1of1 6860 |
. . . 4
⊢ (𝐸:dom 𝐸–1-1-onto→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}) → 𝐸:dom 𝐸–1-1→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
193 | 191, 192 | syl 17 |
. . 3
⊢ ((((({0,
1} ∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5})))) ∧
𝐸 = 〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉) → 𝐸:dom 𝐸–1-1→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}})) |
194 | | 0elfz 13677 |
. . . . . . . . . . 11
⊢ (5 ∈
ℕ0 → 0 ∈ (0...5)) |
195 | 43, 194 | ax-mp 5 |
. . . . . . . . . 10
⊢ 0 ∈
(0...5) |
196 | | 5re 12376 |
. . . . . . . . . . . 12
⊢ 5 ∈
ℝ |
197 | 55, 196, 64 | ltleii 11409 |
. . . . . . . . . . 11
⊢ 1 ≤
5 |
198 | | elfz2nn0 13671 |
. . . . . . . . . . 11
⊢ (1 ∈
(0...5) ↔ (1 ∈ ℕ0 ∧ 5 ∈ ℕ0
∧ 1 ≤ 5)) |
199 | 12, 43, 197, 198 | mpbir3an 1341 |
. . . . . . . . . 10
⊢ 1 ∈
(0...5) |
200 | | prssi 4846 |
. . . . . . . . . 10
⊢ ((0
∈ (0...5) ∧ 1 ∈ (0...5)) → {0, 1} ⊆
(0...5)) |
201 | 195, 199,
200 | mp2an 691 |
. . . . . . . . 9
⊢ {0, 1}
⊆ (0...5) |
202 | 102, 196,
112 | ltleii 11409 |
. . . . . . . . . . 11
⊢ 2 ≤
5 |
203 | | elfz2nn0 13671 |
. . . . . . . . . . 11
⊢ (2 ∈
(0...5) ↔ (2 ∈ ℕ0 ∧ 5 ∈ ℕ0
∧ 2 ≤ 5)) |
204 | 14, 43, 202, 203 | mpbir3an 1341 |
. . . . . . . . . 10
⊢ 2 ∈
(0...5) |
205 | | prssi 4846 |
. . . . . . . . . 10
⊢ ((1
∈ (0...5) ∧ 2 ∈ (0...5)) → {1, 2} ⊆
(0...5)) |
206 | 199, 204,
205 | mp2an 691 |
. . . . . . . . 9
⊢ {1, 2}
⊆ (0...5) |
207 | 133, 196,
136 | ltleii 11409 |
. . . . . . . . . . 11
⊢ 3 ≤
5 |
208 | | elfz2nn0 13671 |
. . . . . . . . . . 11
⊢ (3 ∈
(0...5) ↔ (3 ∈ ℕ0 ∧ 5 ∈ ℕ0
∧ 3 ≤ 5)) |
209 | 23, 43, 207, 208 | mpbir3an 1341 |
. . . . . . . . . 10
⊢ 3 ∈
(0...5) |
210 | | prssi 4846 |
. . . . . . . . . 10
⊢ ((2
∈ (0...5) ∧ 3 ∈ (0...5)) → {2, 3} ⊆
(0...5)) |
211 | 204, 209,
210 | mp2an 691 |
. . . . . . . . 9
⊢ {2, 3}
⊆ (0...5) |
212 | | sseq1 4028 |
. . . . . . . . . 10
⊢ (𝑒 = {0, 1} → (𝑒 ⊆ (0...5) ↔ {0, 1}
⊆ (0...5))) |
213 | | sseq1 4028 |
. . . . . . . . . 10
⊢ (𝑒 = {1, 2} → (𝑒 ⊆ (0...5) ↔ {1, 2}
⊆ (0...5))) |
214 | | sseq1 4028 |
. . . . . . . . . 10
⊢ (𝑒 = {2, 3} → (𝑒 ⊆ (0...5) ↔ {2, 3}
⊆ (0...5))) |
215 | 1, 2, 3, 212, 213, 214 | raltp 4730 |
. . . . . . . . 9
⊢
(∀𝑒 ∈
{{0, 1}, {1, 2}, {2, 3}}𝑒
⊆ (0...5) ↔ ({0, 1} ⊆ (0...5) ∧ {1, 2} ⊆ (0...5)
∧ {2, 3} ⊆ (0...5))) |
216 | 201, 206,
211, 215 | mpbir3an 1341 |
. . . . . . . 8
⊢
∀𝑒 ∈
{{0, 1}, {1, 2}, {2, 3}}𝑒
⊆ (0...5) |
217 | 161, 196,
162 | ltleii 11409 |
. . . . . . . . . . 11
⊢ 4 ≤
5 |
218 | | elfz2nn0 13671 |
. . . . . . . . . . 11
⊢ (4 ∈
(0...5) ↔ (4 ∈ ℕ0 ∧ 5 ∈ ℕ0
∧ 4 ≤ 5)) |
219 | 33, 43, 217, 218 | mpbir3an 1341 |
. . . . . . . . . 10
⊢ 4 ∈
(0...5) |
220 | | prssi 4846 |
. . . . . . . . . 10
⊢ ((3
∈ (0...5) ∧ 4 ∈ (0...5)) → {3, 4} ⊆
(0...5)) |
221 | 209, 219,
220 | mp2an 691 |
. . . . . . . . 9
⊢ {3, 4}
⊆ (0...5) |
222 | | sseq1 4028 |
. . . . . . . . . 10
⊢ (𝑒 = {3, 4} → (𝑒 ⊆ (0...5) ↔ {3, 4}
⊆ (0...5))) |
223 | 5, 222 | ralsn 4705 |
. . . . . . . . 9
⊢
(∀𝑒 ∈
{{3, 4}}𝑒 ⊆ (0...5)
↔ {3, 4} ⊆ (0...5)) |
224 | 221, 223 | mpbir 231 |
. . . . . . . 8
⊢
∀𝑒 ∈
{{3, 4}}𝑒 ⊆
(0...5) |
225 | | ralunb 4214 |
. . . . . . . 8
⊢
(∀𝑒 ∈
({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})𝑒 ⊆ (0...5) ↔ (∀𝑒 ∈ {{0, 1}, {1, 2}, {2,
3}}𝑒 ⊆ (0...5) ∧
∀𝑒 ∈ {{3,
4}}𝑒 ⊆
(0...5))) |
226 | 216, 224,
225 | mpbir2an 710 |
. . . . . . 7
⊢
∀𝑒 ∈
({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})𝑒 ⊆ (0...5) |
227 | 196 | leidi 11820 |
. . . . . . . . . 10
⊢ 5 ≤
5 |
228 | | elfz2nn0 13671 |
. . . . . . . . . 10
⊢ (5 ∈
(0...5) ↔ (5 ∈ ℕ0 ∧ 5 ∈ ℕ0
∧ 5 ≤ 5)) |
229 | 43, 43, 227, 228 | mpbir3an 1341 |
. . . . . . . . 9
⊢ 5 ∈
(0...5) |
230 | | prssi 4846 |
. . . . . . . . 9
⊢ ((4
∈ (0...5) ∧ 5 ∈ (0...5)) → {4, 5} ⊆
(0...5)) |
231 | 219, 229,
230 | mp2an 691 |
. . . . . . . 8
⊢ {4, 5}
⊆ (0...5) |
232 | | prssi 4846 |
. . . . . . . . 9
⊢ ((0
∈ (0...5) ∧ 3 ∈ (0...5)) → {0, 3} ⊆
(0...5)) |
233 | 195, 209,
232 | mp2an 691 |
. . . . . . . 8
⊢ {0, 3}
⊆ (0...5) |
234 | | prssi 4846 |
. . . . . . . . 9
⊢ ((0
∈ (0...5) ∧ 5 ∈ (0...5)) → {0, 5} ⊆
(0...5)) |
235 | 195, 229,
234 | mp2an 691 |
. . . . . . . 8
⊢ {0, 5}
⊆ (0...5) |
236 | | sseq1 4028 |
. . . . . . . . 9
⊢ (𝑒 = {4, 5} → (𝑒 ⊆ (0...5) ↔ {4, 5}
⊆ (0...5))) |
237 | | sseq1 4028 |
. . . . . . . . 9
⊢ (𝑒 = {0, 3} → (𝑒 ⊆ (0...5) ↔ {0, 3}
⊆ (0...5))) |
238 | | sseq1 4028 |
. . . . . . . . 9
⊢ (𝑒 = {0, 5} → (𝑒 ⊆ (0...5) ↔ {0, 5}
⊆ (0...5))) |
239 | 6, 7, 8, 236, 237, 238 | raltp 4730 |
. . . . . . . 8
⊢
(∀𝑒 ∈
{{4, 5}, {0, 3}, {0, 5}}𝑒
⊆ (0...5) ↔ ({4, 5} ⊆ (0...5) ∧ {0, 3} ⊆ (0...5)
∧ {0, 5} ⊆ (0...5))) |
240 | 231, 233,
235, 239 | mpbir3an 1341 |
. . . . . . 7
⊢
∀𝑒 ∈
{{4, 5}, {0, 3}, {0, 5}}𝑒
⊆ (0...5) |
241 | | ralunb 4214 |
. . . . . . 7
⊢
(∀𝑒 ∈
(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}})𝑒 ⊆ (0...5) ↔
(∀𝑒 ∈ ({{0, 1},
{1, 2}, {2, 3}} ∪ {{3, 4}})𝑒 ⊆ (0...5) ∧ ∀𝑒 ∈ {{4, 5}, {0, 3}, {0,
5}}𝑒 ⊆
(0...5))) |
242 | 226, 240,
241 | mpbir2an 710 |
. . . . . 6
⊢
∀𝑒 ∈
(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}})𝑒 ⊆
(0...5) |
243 | | pwssb 5127 |
. . . . . 6
⊢ ((({{0,
1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆
𝒫 (0...5) ↔ ∀𝑒 ∈ (({{0, 1}, {1, 2}, {2, 3}} ∪ {{3,
4}}) ∪ {{4, 5}, {0, 3}, {0, 5}})𝑒 ⊆ (0...5)) |
244 | 242, 243 | mpbir 231 |
. . . . 5
⊢ (({{0,
1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆
𝒫 (0...5) |
245 | | usgrexmpl2.v |
. . . . . 6
⊢ 𝑉 = (0...5) |
246 | 245 | pweqi 4638 |
. . . . 5
⊢ 𝒫
𝑉 = 𝒫
(0...5) |
247 | 244, 246 | sseqtrri 4040 |
. . . 4
⊢ (({{0,
1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆
𝒫 𝑉 |
248 | | prhash2ex 14444 |
. . . . . . 7
⊢
(♯‘{0, 1}) = 2 |
249 | | 1ex 11282 |
. . . . . . . . 9
⊢ 1 ∈
V |
250 | | 2ex 12366 |
. . . . . . . . 9
⊢ 2 ∈
V |
251 | 249, 250,
73 | 3pm3.2i 1339 |
. . . . . . . 8
⊢ (1 ∈
V ∧ 2 ∈ V ∧ 1 ≠ 2) |
252 | | hashprb 14442 |
. . . . . . . 8
⊢ ((1
∈ V ∧ 2 ∈ V ∧ 1 ≠ 2) ↔ (♯‘{1, 2}) =
2) |
253 | 251, 252 | mpbi 230 |
. . . . . . 7
⊢
(♯‘{1, 2}) = 2 |
254 | | 3ex 12371 |
. . . . . . . . 9
⊢ 3 ∈
V |
255 | 250, 254,
104 | 3pm3.2i 1339 |
. . . . . . . 8
⊢ (2 ∈
V ∧ 3 ∈ V ∧ 2 ≠ 3) |
256 | | hashprb 14442 |
. . . . . . . 8
⊢ ((2
∈ V ∧ 3 ∈ V ∧ 2 ≠ 3) ↔ (♯‘{2, 3}) =
2) |
257 | 255, 256 | mpbi 230 |
. . . . . . 7
⊢
(♯‘{2, 3}) = 2 |
258 | | fveqeq2 6928 |
. . . . . . . 8
⊢ (𝑒 = {0, 1} →
((♯‘𝑒) = 2
↔ (♯‘{0, 1}) = 2)) |
259 | | fveqeq2 6928 |
. . . . . . . 8
⊢ (𝑒 = {1, 2} →
((♯‘𝑒) = 2
↔ (♯‘{1, 2}) = 2)) |
260 | | fveqeq2 6928 |
. . . . . . . 8
⊢ (𝑒 = {2, 3} →
((♯‘𝑒) = 2
↔ (♯‘{2, 3}) = 2)) |
261 | 1, 2, 3, 258, 259, 260 | raltp 4730 |
. . . . . . 7
⊢
(∀𝑒 ∈
{{0, 1}, {1, 2}, {2, 3}} (♯‘𝑒) = 2 ↔ ((♯‘{0, 1}) = 2
∧ (♯‘{1, 2}) = 2 ∧ (♯‘{2, 3}) =
2)) |
262 | 248, 253,
257, 261 | mpbir3an 1341 |
. . . . . 6
⊢
∀𝑒 ∈
{{0, 1}, {1, 2}, {2, 3}} (♯‘𝑒) = 2 |
263 | 33 | elexi 3506 |
. . . . . . . . 9
⊢ 4 ∈
V |
264 | 254, 263,
135 | 3pm3.2i 1339 |
. . . . . . . 8
⊢ (3 ∈
V ∧ 4 ∈ V ∧ 3 ≠ 4) |
265 | | hashprb 14442 |
. . . . . . . 8
⊢ ((3
∈ V ∧ 4 ∈ V ∧ 3 ≠ 4) ↔ (♯‘{3, 4}) =
2) |
266 | 264, 265 | mpbi 230 |
. . . . . . 7
⊢
(♯‘{3, 4}) = 2 |
267 | | fveqeq2 6928 |
. . . . . . . 8
⊢ (𝑒 = {3, 4} →
((♯‘𝑒) = 2
↔ (♯‘{3, 4}) = 2)) |
268 | 5, 267 | ralsn 4705 |
. . . . . . 7
⊢
(∀𝑒 ∈
{{3, 4}} (♯‘𝑒)
= 2 ↔ (♯‘{3, 4}) = 2) |
269 | 266, 268 | mpbir 231 |
. . . . . 6
⊢
∀𝑒 ∈
{{3, 4}} (♯‘𝑒)
= 2 |
270 | | ralunb 4214 |
. . . . . 6
⊢
(∀𝑒 ∈
({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})(♯‘𝑒) = 2 ↔ (∀𝑒 ∈ {{0, 1}, {1, 2}, {2, 3}}
(♯‘𝑒) = 2 ∧
∀𝑒 ∈ {{3, 4}}
(♯‘𝑒) =
2)) |
271 | 262, 269,
270 | mpbir2an 710 |
. . . . 5
⊢
∀𝑒 ∈
({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})(♯‘𝑒) = 2 |
272 | 43 | elexi 3506 |
. . . . . . . 8
⊢ 5 ∈
V |
273 | 263, 272,
163 | 3pm3.2i 1339 |
. . . . . . 7
⊢ (4 ∈
V ∧ 5 ∈ V ∧ 4 ≠ 5) |
274 | | hashprb 14442 |
. . . . . . 7
⊢ ((4
∈ V ∧ 5 ∈ V ∧ 4 ≠ 5) ↔ (♯‘{4, 5}) =
2) |
275 | 273, 274 | mpbi 230 |
. . . . . 6
⊢
(♯‘{4, 5}) = 2 |
276 | | c0ex 11280 |
. . . . . . . 8
⊢ 0 ∈
V |
277 | 276, 254,
28 | 3pm3.2i 1339 |
. . . . . . 7
⊢ (0 ∈
V ∧ 3 ∈ V ∧ 0 ≠ 3) |
278 | | hashprb 14442 |
. . . . . . 7
⊢ ((0
∈ V ∧ 3 ∈ V ∧ 0 ≠ 3) ↔ (♯‘{0, 3}) =
2) |
279 | 277, 278 | mpbi 230 |
. . . . . 6
⊢
(♯‘{0, 3}) = 2 |
280 | 276, 272,
47 | 3pm3.2i 1339 |
. . . . . . 7
⊢ (0 ∈
V ∧ 5 ∈ V ∧ 0 ≠ 5) |
281 | | hashprb 14442 |
. . . . . . 7
⊢ ((0
∈ V ∧ 5 ∈ V ∧ 0 ≠ 5) ↔ (♯‘{0, 5}) =
2) |
282 | 280, 281 | mpbi 230 |
. . . . . 6
⊢
(♯‘{0, 5}) = 2 |
283 | | fveqeq2 6928 |
. . . . . . 7
⊢ (𝑒 = {4, 5} →
((♯‘𝑒) = 2
↔ (♯‘{4, 5}) = 2)) |
284 | | fveqeq2 6928 |
. . . . . . 7
⊢ (𝑒 = {0, 3} →
((♯‘𝑒) = 2
↔ (♯‘{0, 3}) = 2)) |
285 | | fveqeq2 6928 |
. . . . . . 7
⊢ (𝑒 = {0, 5} →
((♯‘𝑒) = 2
↔ (♯‘{0, 5}) = 2)) |
286 | 6, 7, 8, 283, 284, 285 | raltp 4730 |
. . . . . 6
⊢
(∀𝑒 ∈
{{4, 5}, {0, 3}, {0, 5}} (♯‘𝑒) = 2 ↔ ((♯‘{4, 5}) = 2
∧ (♯‘{0, 3}) = 2 ∧ (♯‘{0, 5}) =
2)) |
287 | 275, 279,
282, 286 | mpbir3an 1341 |
. . . . 5
⊢
∀𝑒 ∈
{{4, 5}, {0, 3}, {0, 5}} (♯‘𝑒) = 2 |
288 | | ralunb 4214 |
. . . . 5
⊢
(∀𝑒 ∈
(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0,
5}})(♯‘𝑒) = 2
↔ (∀𝑒 ∈
({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})(♯‘𝑒) = 2 ∧ ∀𝑒 ∈ {{4, 5}, {0, 3}, {0, 5}}
(♯‘𝑒) =
2)) |
289 | 271, 287,
288 | mpbir2an 710 |
. . . 4
⊢
∀𝑒 ∈
(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0,
5}})(♯‘𝑒) =
2 |
290 | | ssrab 4090 |
. . . 4
⊢ ((({{0,
1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆
{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} ↔ ((({{0, 1}, {1,
2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆ 𝒫
𝑉 ∧ ∀𝑒 ∈ (({{0, 1}, {1, 2}, {2,
3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}})(♯‘𝑒) = 2)) |
291 | 247, 289,
290 | mpbir2an 710 |
. . 3
⊢ (({{0,
1}, {1, 2}, {2, 3}} ∪ {{3, 4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆
{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |
292 | | f1ss 6821 |
. . 3
⊢ ((𝐸:dom 𝐸–1-1→(({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}})
∪ {{4, 5}, {0, 3}, {0, 5}}) ∧ (({{0, 1}, {1, 2}, {2, 3}} ∪ {{3,
4}}) ∪ {{4, 5}, {0, 3}, {0, 5}}) ⊆ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) → 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
293 | 193, 291,
292 | sylancl 585 |
. 2
⊢ ((((({0,
1} ∈ V ∧ {1, 2} ∈ V ∧ {2, 3} ∈ V) ∧ {3, 4} ∈ V
∧ ({4, 5} ∈ V ∧ {0, 3} ∈ V ∧ {0, 5} ∈ V)) ∧
(((({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 3} ∧ {0, 1} ≠ {3, 4}) ∧
({0, 1} ≠ {4, 5} ∧ {0, 1} ≠ {0, 3} ∧ {0, 1} ≠ {0, 5})) ∧
(({1, 2} ≠ {2, 3} ∧ {1, 2} ≠ {3, 4}) ∧ ({1, 2} ≠ {4, 5} ∧
{1, 2} ≠ {0, 3} ∧ {1, 2} ≠ {0, 5})) ∧ ({2, 3} ≠ {3, 4} ∧
({2, 3} ≠ {4, 5} ∧ {2, 3} ≠ {0, 3} ∧ {2, 3} ≠ {0, 5}))) ∧
(({3, 4} ≠ {4, 5} ∧ {3, 4} ≠ {0, 3} ∧ {3, 4} ≠ {0, 5}) ∧
({4, 5} ≠ {0, 3} ∧ {4, 5} ≠ {0, 5} ∧ {0, 3} ≠ {0, 5})))) ∧
𝐸 = 〈“{0, 1} {1,
2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”〉) → 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) |
294 | 175, 176,
293 | mp2an 691 |
1
⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} |