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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 2100 | . . . 4 | |
2 | 1 | cbvexv 1795 | . . 3 |
3 | eleq1 2100 | . . . 4 | |
4 | 3 | cbvexv 1795 | . . 3 |
5 | 2, 4 | bitri 173 | . 2 |
6 | r19.2m 3309 | . . . . 5 | |
7 | errel 6115 | . . . . . . 7 | |
8 | df-rel 4352 | . . . . . . 7 | |
9 | 7, 8 | sylib 127 | . . . . . 6 |
10 | 9 | reximi 2416 | . . . . 5 |
11 | iinss 3708 | . . . . 5 | |
12 | 6, 10, 11 | 3syl 17 | . . . 4 |
13 | df-rel 4352 | . . . 4 | |
14 | 12, 13 | sylibr 137 | . . 3 |
15 | id 19 | . . . . . . . . . 10 | |
16 | 15 | ersymb 6120 | . . . . . . . . 9 |
17 | 16 | biimpd 132 | . . . . . . . 8 |
18 | df-br 3765 | . . . . . . . 8 | |
19 | df-br 3765 | . . . . . . . 8 | |
20 | 17, 18, 19 | 3imtr3g 193 | . . . . . . 7 |
21 | 20 | ral2imi 2385 | . . . . . 6 |
22 | 21 | adantl 262 | . . . . 5 |
23 | df-br 3765 | . . . . . 6 | |
24 | vex 2560 | . . . . . . . 8 | |
25 | vex 2560 | . . . . . . . 8 | |
26 | 24, 25 | opex 3966 | . . . . . . 7 |
27 | eliin 3662 | . . . . . . 7 | |
28 | 26, 27 | ax-mp 7 | . . . . . 6 |
29 | 23, 28 | bitri 173 | . . . . 5 |
30 | df-br 3765 | . . . . . 6 | |
31 | 25, 24 | opex 3966 | . . . . . . 7 |
32 | eliin 3662 | . . . . . . 7 | |
33 | 31, 32 | ax-mp 7 | . . . . . 6 |
34 | 30, 33 | bitri 173 | . . . . 5 |
35 | 22, 29, 34 | 3imtr4g 194 | . . . 4 |
36 | 35 | imp 115 | . . 3 |
37 | r19.26 2441 | . . . . . 6 | |
38 | 15 | ertr 6121 | . . . . . . . . 9 |
39 | df-br 3765 | . . . . . . . . . 10 | |
40 | 18, 39 | anbi12i 433 | . . . . . . . . 9 |
41 | df-br 3765 | . . . . . . . . 9 | |
42 | 38, 40, 41 | 3imtr3g 193 | . . . . . . . 8 |
43 | 42 | ral2imi 2385 | . . . . . . 7 |
44 | 43 | adantl 262 | . . . . . 6 |
45 | 37, 44 | syl5bir 142 | . . . . 5 |
46 | df-br 3765 | . . . . . . 7 | |
47 | vex 2560 | . . . . . . . . 9 | |
48 | 25, 47 | opex 3966 | . . . . . . . 8 |
49 | eliin 3662 | . . . . . . . 8 | |
50 | 48, 49 | ax-mp 7 | . . . . . . 7 |
51 | 46, 50 | bitri 173 | . . . . . 6 |
52 | 29, 51 | anbi12i 433 | . . . . 5 |
53 | df-br 3765 | . . . . . 6 | |
54 | 24, 47 | opex 3966 | . . . . . . 7 |
55 | eliin 3662 | . . . . . . 7 | |
56 | 54, 55 | ax-mp 7 | . . . . . 6 |
57 | 53, 56 | bitri 173 | . . . . 5 |
58 | 45, 52, 57 | 3imtr4g 194 | . . . 4 |
59 | 58 | imp 115 | . . 3 |
60 | simpl 102 | . . . . . . . . . . 11 | |
61 | simpr 103 | . . . . . . . . . . 11 | |
62 | 60, 61 | erref 6126 | . . . . . . . . . 10 |
63 | df-br 3765 | . . . . . . . . . 10 | |
64 | 62, 63 | sylib 127 | . . . . . . . . 9 |
65 | 64 | expcom 109 | . . . . . . . 8 |
66 | 65 | ralimdv 2388 | . . . . . . 7 |
67 | 66 | com12 27 | . . . . . 6 |
68 | 67 | adantl 262 | . . . . 5 |
69 | r19.26 2441 | . . . . . . 7 | |
70 | r19.2m 3309 | . . . . . . . . 9 | |
71 | 24, 24 | opeldm 4538 | . . . . . . . . . . 11 |
72 | erdm 6116 | . . . . . . . . . . . . 13 | |
73 | 72 | eleq2d 2107 | . . . . . . . . . . . 12 |
74 | 73 | biimpa 280 | . . . . . . . . . . 11 |
75 | 71, 74 | sylan2 270 | . . . . . . . . . 10 |
76 | 75 | rexlimivw 2429 | . . . . . . . . 9 |
77 | 70, 76 | syl 14 | . . . . . . . 8 |
78 | 77 | ex 108 | . . . . . . 7 |
79 | 69, 78 | syl5bir 142 | . . . . . 6 |
80 | 79 | expdimp 246 | . . . . 5 |
81 | 68, 80 | impbid 120 | . . . 4 |
82 | df-br 3765 | . . . . 5 | |
83 | 24, 24 | opex 3966 | . . . . . 6 |
84 | eliin 3662 | . . . . . 6 | |
85 | 83, 84 | ax-mp 7 | . . . . 5 |
86 | 82, 85 | bitri 173 | . . . 4 |
87 | 81, 86 | syl6bbr 187 | . . 3 |
88 | 14, 36, 59, 87 | iserd 6132 | . 2 |
89 | 5, 88 | sylanbr 269 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wex 1381 wcel 1393 wral 2306 wrex 2307 cvv 2557 wss 2917 cop 3378 ciin 3658 class class class wbr 3764 cxp 4343 cdm 4345 wrel 4350 wer 6103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-iin 3660 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-er 6106 |
This theorem is referenced by: riinerm 6179 |
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