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Mirrors > Home > ILE Home > Th. List > erinxp | Unicode version |
Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
erinxp.r | |
erinxp.a |
Ref | Expression |
---|---|
erinxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 3267 | . . . 4 | |
2 | relxp 4618 | . . . 4 | |
3 | relss 4596 | . . . 4 | |
4 | 1, 2, 3 | mp2 16 | . . 3 |
5 | 4 | a1i 9 | . 2 |
6 | simpr 109 | . . . . 5 | |
7 | brinxp2 4576 | . . . . 5 | |
8 | 6, 7 | sylib 121 | . . . 4 |
9 | 8 | simp2d 979 | . . 3 |
10 | 8 | simp1d 978 | . . 3 |
11 | erinxp.r | . . . . 5 | |
12 | 11 | adantr 274 | . . . 4 |
13 | 8 | simp3d 980 | . . . 4 |
14 | 12, 13 | ersym 6409 | . . 3 |
15 | brinxp2 4576 | . . 3 | |
16 | 9, 10, 14, 15 | syl3anbrc 1150 | . 2 |
17 | 10 | adantrr 470 | . . 3 |
18 | simprr 506 | . . . . 5 | |
19 | brinxp2 4576 | . . . . 5 | |
20 | 18, 19 | sylib 121 | . . . 4 |
21 | 20 | simp2d 979 | . . 3 |
22 | 11 | adantr 274 | . . . 4 |
23 | 13 | adantrr 470 | . . . 4 |
24 | 20 | simp3d 980 | . . . 4 |
25 | 22, 23, 24 | ertrd 6413 | . . 3 |
26 | brinxp2 4576 | . . 3 | |
27 | 17, 21, 25, 26 | syl3anbrc 1150 | . 2 |
28 | 11 | adantr 274 | . . . . . 6 |
29 | erinxp.a | . . . . . . 7 | |
30 | 29 | sselda 3067 | . . . . . 6 |
31 | 28, 30 | erref 6417 | . . . . 5 |
32 | 31 | ex 114 | . . . 4 |
33 | 32 | pm4.71rd 391 | . . 3 |
34 | brin 3950 | . . . 4 | |
35 | brxp 4540 | . . . . . 6 | |
36 | anidm 393 | . . . . . 6 | |
37 | 35, 36 | bitri 183 | . . . . 5 |
38 | 37 | anbi2i 452 | . . . 4 |
39 | 34, 38 | bitri 183 | . . 3 |
40 | 33, 39 | syl6bbr 197 | . 2 |
41 | 5, 16, 27, 40 | iserd 6423 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wcel 1465 cin 3040 wss 3041 class class class wbr 3899 cxp 4507 wrel 4514 wer 6394 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-er 6397 |
This theorem is referenced by: (None) |
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