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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 2198 | . . . 4 | |
2 | 1 | cbvexv 1890 | . . 3 |
3 | eleq1w 2198 | . . . 4 | |
4 | 3 | cbvexv 1890 | . . 3 |
5 | 2, 4 | bitri 183 | . 2 |
6 | r19.2m 3444 | . . . . 5 | |
7 | errel 6431 | . . . . . . 7 | |
8 | df-rel 4541 | . . . . . . 7 | |
9 | 7, 8 | sylib 121 | . . . . . 6 |
10 | 9 | reximi 2527 | . . . . 5 |
11 | iinss 3859 | . . . . 5 | |
12 | 6, 10, 11 | 3syl 17 | . . . 4 |
13 | df-rel 4541 | . . . 4 | |
14 | 12, 13 | sylibr 133 | . . 3 |
15 | id 19 | . . . . . . . . . 10 | |
16 | 15 | ersymb 6436 | . . . . . . . . 9 |
17 | 16 | biimpd 143 | . . . . . . . 8 |
18 | df-br 3925 | . . . . . . . 8 | |
19 | df-br 3925 | . . . . . . . 8 | |
20 | 17, 18, 19 | 3imtr3g 203 | . . . . . . 7 |
21 | 20 | ral2imi 2495 | . . . . . 6 |
22 | 21 | adantl 275 | . . . . 5 |
23 | df-br 3925 | . . . . . 6 | |
24 | vex 2684 | . . . . . . . 8 | |
25 | vex 2684 | . . . . . . . 8 | |
26 | 24, 25 | opex 4146 | . . . . . . 7 |
27 | eliin 3813 | . . . . . . 7 | |
28 | 26, 27 | ax-mp 5 | . . . . . 6 |
29 | 23, 28 | bitri 183 | . . . . 5 |
30 | df-br 3925 | . . . . . 6 | |
31 | 25, 24 | opex 4146 | . . . . . . 7 |
32 | eliin 3813 | . . . . . . 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 2556 | . . . . . 6 | |
38 | 15 | ertr 6437 | . . . . . . . . 9 |
39 | df-br 3925 | . . . . . . . . . 10 | |
40 | 18, 39 | anbi12i 455 | . . . . . . . . 9 |
41 | df-br 3925 | . . . . . . . . 9 | |
42 | 38, 40, 41 | 3imtr3g 203 | . . . . . . . 8 |
43 | 42 | ral2imi 2495 | . . . . . . 7 |
44 | 43 | adantl 275 | . . . . . 6 |
45 | 37, 44 | syl5bir 152 | . . . . 5 |
46 | df-br 3925 | . . . . . . 7 | |
47 | vex 2684 | . . . . . . . . 9 | |
48 | 25, 47 | opex 4146 | . . . . . . . 8 |
49 | eliin 3813 | . . . . . . . 8 | |
50 | 48, 49 | ax-mp 5 | . . . . . . 7 |
51 | 46, 50 | bitri 183 | . . . . . 6 |
52 | 29, 51 | anbi12i 455 | . . . . 5 |
53 | df-br 3925 | . . . . . 6 | |
54 | 24, 47 | opex 4146 | . . . . . . 7 |
55 | eliin 3813 | . . . . . . 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 6442 | . . . . . . . . . 10 |
63 | df-br 3925 | . . . . . . . . . 10 | |
64 | 62, 63 | sylib 121 | . . . . . . . . 9 |
65 | 64 | expcom 115 | . . . . . . . 8 |
66 | 65 | ralimdv 2498 | . . . . . . 7 |
67 | 66 | com12 30 | . . . . . 6 |
68 | 67 | adantl 275 | . . . . 5 |
69 | r19.26 2556 | . . . . . . 7 | |
70 | r19.2m 3444 | . . . . . . . . 9 | |
71 | 24, 24 | opeldm 4737 | . . . . . . . . . . 11 |
72 | erdm 6432 | . . . . . . . . . . . . 13 | |
73 | 72 | eleq2d 2207 | . . . . . . . . . . . 12 |
74 | 73 | biimpa 294 | . . . . . . . . . . 11 |
75 | 71, 74 | sylan2 284 | . . . . . . . . . 10 |
76 | 75 | rexlimivw 2543 | . . . . . . . . 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 3925 | . . . . 5 | |
83 | 24, 24 | opex 4146 | . . . . . 6 |
84 | eliin 3813 | . . . . . 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 6448 | . 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 2414 wrex 2415 cvv 2681 wss 3066 cop 3525 ciin 3809 class class class wbr 3924 cxp 4532 cdm 4534 wrel 4539 wer 6419 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-iin 3811 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-er 6422 |
This theorem is referenced by: riinerm 6495 |
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