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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 | eleq1w 2200 | . . . 4 | |
2 | 1 | cbvexv 1890 | . . 3 |
3 | eleq1w 2200 | . . . 4 | |
4 | 3 | cbvexv 1890 | . . 3 |
5 | 2, 4 | bitri 183 | . 2 |
6 | r19.2m 3449 | . . . . 5 | |
7 | errel 6438 | . . . . . . 7 | |
8 | df-rel 4546 | . . . . . . 7 | |
9 | 7, 8 | sylib 121 | . . . . . 6 |
10 | 9 | reximi 2529 | . . . . 5 |
11 | iinss 3864 | . . . . 5 | |
12 | 6, 10, 11 | 3syl 17 | . . . 4 |
13 | df-rel 4546 | . . . 4 | |
14 | 12, 13 | sylibr 133 | . . 3 |
15 | id 19 | . . . . . . . . . 10 | |
16 | 15 | ersymb 6443 | . . . . . . . . 9 |
17 | 16 | biimpd 143 | . . . . . . . 8 |
18 | df-br 3930 | . . . . . . . 8 | |
19 | df-br 3930 | . . . . . . . 8 | |
20 | 17, 18, 19 | 3imtr3g 203 | . . . . . . 7 |
21 | 20 | ral2imi 2497 | . . . . . 6 |
22 | 21 | adantl 275 | . . . . 5 |
23 | df-br 3930 | . . . . . 6 | |
24 | vex 2689 | . . . . . . . 8 | |
25 | vex 2689 | . . . . . . . 8 | |
26 | 24, 25 | opex 4151 | . . . . . . 7 |
27 | eliin 3818 | . . . . . . 7 | |
28 | 26, 27 | ax-mp 5 | . . . . . 6 |
29 | 23, 28 | bitri 183 | . . . . 5 |
30 | df-br 3930 | . . . . . 6 | |
31 | 25, 24 | opex 4151 | . . . . . . 7 |
32 | eliin 3818 | . . . . . . 7 | |
33 | 31, 32 | ax-mp 5 | . . . . . 6 |
34 | 30, 33 | bitri 183 | . . . . 5 |
35 | 22, 29, 34 | 3imtr4g 204 | . . . 4 |
36 | 35 | imp 123 | . . 3 |
37 | r19.26 2558 | . . . . . 6 | |
38 | 15 | ertr 6444 | . . . . . . . . 9 |
39 | df-br 3930 | . . . . . . . . . 10 | |
40 | 18, 39 | anbi12i 455 | . . . . . . . . 9 |
41 | df-br 3930 | . . . . . . . . 9 | |
42 | 38, 40, 41 | 3imtr3g 203 | . . . . . . . 8 |
43 | 42 | ral2imi 2497 | . . . . . . 7 |
44 | 43 | adantl 275 | . . . . . 6 |
45 | 37, 44 | syl5bir 152 | . . . . 5 |
46 | df-br 3930 | . . . . . . 7 | |
47 | vex 2689 | . . . . . . . . 9 | |
48 | 25, 47 | opex 4151 | . . . . . . . 8 |
49 | eliin 3818 | . . . . . . . 8 | |
50 | 48, 49 | ax-mp 5 | . . . . . . 7 |
51 | 46, 50 | bitri 183 | . . . . . 6 |
52 | 29, 51 | anbi12i 455 | . . . . 5 |
53 | df-br 3930 | . . . . . 6 | |
54 | 24, 47 | opex 4151 | . . . . . . 7 |
55 | eliin 3818 | . . . . . . 7 | |
56 | 54, 55 | ax-mp 5 | . . . . . 6 |
57 | 53, 56 | bitri 183 | . . . . 5 |
58 | 45, 52, 57 | 3imtr4g 204 | . . . 4 |
59 | 58 | imp 123 | . . 3 |
60 | simpl 108 | . . . . . . . . . . 11 | |
61 | simpr 109 | . . . . . . . . . . 11 | |
62 | 60, 61 | erref 6449 | . . . . . . . . . 10 |
63 | df-br 3930 | . . . . . . . . . 10 | |
64 | 62, 63 | sylib 121 | . . . . . . . . 9 |
65 | 64 | expcom 115 | . . . . . . . 8 |
66 | 65 | ralimdv 2500 | . . . . . . 7 |
67 | 66 | com12 30 | . . . . . 6 |
68 | 67 | adantl 275 | . . . . 5 |
69 | r19.26 2558 | . . . . . . 7 | |
70 | r19.2m 3449 | . . . . . . . . 9 | |
71 | 24, 24 | opeldm 4742 | . . . . . . . . . . 11 |
72 | erdm 6439 | . . . . . . . . . . . . 13 | |
73 | 72 | eleq2d 2209 | . . . . . . . . . . . 12 |
74 | 73 | biimpa 294 | . . . . . . . . . . 11 |
75 | 71, 74 | sylan2 284 | . . . . . . . . . 10 |
76 | 75 | rexlimivw 2545 | . . . . . . . . 9 |
77 | 70, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | ex 114 | . . . . . . 7 |
79 | 69, 78 | syl5bir 152 | . . . . . 6 |
80 | 79 | expdimp 257 | . . . . 5 |
81 | 68, 80 | impbid 128 | . . . 4 |
82 | df-br 3930 | . . . . 5 | |
83 | 24, 24 | opex 4151 | . . . . . 6 |
84 | eliin 3818 | . . . . . 6 | |
85 | 83, 84 | ax-mp 5 | . . . . 5 |
86 | 82, 85 | bitri 183 | . . . 4 |
87 | 81, 86 | syl6bbr 197 | . . 3 |
88 | 14, 36, 59, 87 | iserd 6455 | . 2 |
89 | 5, 88 | sylanbr 283 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wex 1468 wcel 1480 wral 2416 wrex 2417 cvv 2686 wss 3071 cop 3530 ciin 3814 class class class wbr 3929 cxp 4537 cdm 4539 wrel 4544 wer 6426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-iin 3816 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-er 6429 |
This theorem is referenced by: riinerm 6502 |
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