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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 3297 | . . . 4 | |
2 | relxp 4648 | . . . 4 | |
3 | relss 4626 | . . . 4 | |
4 | 1, 2, 3 | mp2 16 | . . 3 |
5 | 4 | a1i 9 | . 2 |
6 | simpr 109 | . . . . 5 | |
7 | brinxp2 4606 | . . . . 5 | |
8 | 6, 7 | sylib 121 | . . . 4 |
9 | 8 | simp2d 994 | . . 3 |
10 | 8 | simp1d 993 | . . 3 |
11 | erinxp.r | . . . . 5 | |
12 | 11 | adantr 274 | . . . 4 |
13 | 8 | simp3d 995 | . . . 4 |
14 | 12, 13 | ersym 6441 | . . 3 |
15 | brinxp2 4606 | . . 3 | |
16 | 9, 10, 14, 15 | syl3anbrc 1165 | . 2 |
17 | 10 | adantrr 470 | . . 3 |
18 | simprr 521 | . . . . 5 | |
19 | brinxp2 4606 | . . . . 5 | |
20 | 18, 19 | sylib 121 | . . . 4 |
21 | 20 | simp2d 994 | . . 3 |
22 | 11 | adantr 274 | . . . 4 |
23 | 13 | adantrr 470 | . . . 4 |
24 | 20 | simp3d 995 | . . . 4 |
25 | 22, 23, 24 | ertrd 6445 | . . 3 |
26 | brinxp2 4606 | . . 3 | |
27 | 17, 21, 25, 26 | syl3anbrc 1165 | . 2 |
28 | 11 | adantr 274 | . . . . . 6 |
29 | erinxp.a | . . . . . . 7 | |
30 | 29 | sselda 3097 | . . . . . 6 |
31 | 28, 30 | erref 6449 | . . . . 5 |
32 | 31 | ex 114 | . . . 4 |
33 | 32 | pm4.71rd 391 | . . 3 |
34 | brin 3980 | . . . 4 | |
35 | brxp 4570 | . . . . . 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 6455 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wcel 1480 cin 3070 wss 3071 class class class wbr 3929 cxp 4537 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-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: (None) |
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