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Mirrors > Home > ILE Home > Th. List > iinerm | Unicode version |
Description: The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
iinerm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2141 | . . . 4 | |
2 | 1 | cbvexv 1836 | . . 3 |
3 | eleq1 2141 | . . . 4 | |
4 | 3 | cbvexv 1836 | . . 3 |
5 | 2, 4 | bitri 182 | . 2 |
6 | r19.2m 3329 | . . . . 5 | |
7 | errel 6138 | . . . . . . 7 | |
8 | df-rel 4370 | . . . . . . 7 | |
9 | 7, 8 | sylib 120 | . . . . . 6 |
10 | 9 | reximi 2458 | . . . . 5 |
11 | iinss 3729 | . . . . 5 | |
12 | 6, 10, 11 | 3syl 17 | . . . 4 |
13 | df-rel 4370 | . . . 4 | |
14 | 12, 13 | sylibr 132 | . . 3 |
15 | id 19 | . . . . . . . . . 10 | |
16 | 15 | ersymb 6143 | . . . . . . . . 9 |
17 | 16 | biimpd 142 | . . . . . . . 8 |
18 | df-br 3786 | . . . . . . . 8 | |
19 | df-br 3786 | . . . . . . . 8 | |
20 | 17, 18, 19 | 3imtr3g 202 | . . . . . . 7 |
21 | 20 | ral2imi 2427 | . . . . . 6 |
22 | 21 | adantl 271 | . . . . 5 |
23 | df-br 3786 | . . . . . 6 | |
24 | vex 2604 | . . . . . . . 8 | |
25 | vex 2604 | . . . . . . . 8 | |
26 | 24, 25 | opex 3984 | . . . . . . 7 |
27 | eliin 3683 | . . . . . . 7 | |
28 | 26, 27 | ax-mp 7 | . . . . . 6 |
29 | 23, 28 | bitri 182 | . . . . 5 |
30 | df-br 3786 | . . . . . 6 | |
31 | 25, 24 | opex 3984 | . . . . . . 7 |
32 | eliin 3683 | . . . . . . 7 | |
33 | 31, 32 | ax-mp 7 | . . . . . 6 |
34 | 30, 33 | bitri 182 | . . . . 5 |
35 | 22, 29, 34 | 3imtr4g 203 | . . . 4 |
36 | 35 | imp 122 | . . 3 |
37 | r19.26 2485 | . . . . . 6 | |
38 | 15 | ertr 6144 | . . . . . . . . 9 |
39 | df-br 3786 | . . . . . . . . . 10 | |
40 | 18, 39 | anbi12i 447 | . . . . . . . . 9 |
41 | df-br 3786 | . . . . . . . . 9 | |
42 | 38, 40, 41 | 3imtr3g 202 | . . . . . . . 8 |
43 | 42 | ral2imi 2427 | . . . . . . 7 |
44 | 43 | adantl 271 | . . . . . 6 |
45 | 37, 44 | syl5bir 151 | . . . . 5 |
46 | df-br 3786 | . . . . . . 7 | |
47 | vex 2604 | . . . . . . . . 9 | |
48 | 25, 47 | opex 3984 | . . . . . . . 8 |
49 | eliin 3683 | . . . . . . . 8 | |
50 | 48, 49 | ax-mp 7 | . . . . . . 7 |
51 | 46, 50 | bitri 182 | . . . . . 6 |
52 | 29, 51 | anbi12i 447 | . . . . 5 |
53 | df-br 3786 | . . . . . 6 | |
54 | 24, 47 | opex 3984 | . . . . . . 7 |
55 | eliin 3683 | . . . . . . 7 | |
56 | 54, 55 | ax-mp 7 | . . . . . 6 |
57 | 53, 56 | bitri 182 | . . . . 5 |
58 | 45, 52, 57 | 3imtr4g 203 | . . . 4 |
59 | 58 | imp 122 | . . 3 |
60 | simpl 107 | . . . . . . . . . . 11 | |
61 | simpr 108 | . . . . . . . . . . 11 | |
62 | 60, 61 | erref 6149 | . . . . . . . . . 10 |
63 | df-br 3786 | . . . . . . . . . 10 | |
64 | 62, 63 | sylib 120 | . . . . . . . . 9 |
65 | 64 | expcom 114 | . . . . . . . 8 |
66 | 65 | ralimdv 2430 | . . . . . . 7 |
67 | 66 | com12 30 | . . . . . 6 |
68 | 67 | adantl 271 | . . . . 5 |
69 | r19.26 2485 | . . . . . . 7 | |
70 | r19.2m 3329 | . . . . . . . . 9 | |
71 | 24, 24 | opeldm 4556 | . . . . . . . . . . 11 |
72 | erdm 6139 | . . . . . . . . . . . . 13 | |
73 | 72 | eleq2d 2148 | . . . . . . . . . . . 12 |
74 | 73 | biimpa 290 | . . . . . . . . . . 11 |
75 | 71, 74 | sylan2 280 | . . . . . . . . . 10 |
76 | 75 | rexlimivw 2473 | . . . . . . . . 9 |
77 | 70, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | ex 113 | . . . . . . 7 |
79 | 69, 78 | syl5bir 151 | . . . . . 6 |
80 | 79 | expdimp 255 | . . . . 5 |
81 | 68, 80 | impbid 127 | . . . 4 |
82 | df-br 3786 | . . . . 5 | |
83 | 24, 24 | opex 3984 | . . . . . 6 |
84 | eliin 3683 | . . . . . 6 | |
85 | 83, 84 | ax-mp 7 | . . . . 5 |
86 | 82, 85 | bitri 182 | . . . 4 |
87 | 81, 86 | syl6bbr 196 | . . 3 |
88 | 14, 36, 59, 87 | iserd 6155 | . 2 |
89 | 5, 88 | sylanbr 279 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wex 1421 wcel 1433 wral 2348 wrex 2349 cvv 2601 wss 2973 cop 3401 ciin 3679 class class class wbr 3785 cxp 4361 cdm 4363 wrel 4368 wer 6126 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-iin 3681 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-er 6129 |
This theorem is referenced by: riinerm 6202 |
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