Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3158 | . . . 4 | |
2 | relxp 4447 | . . . 4 | |
3 | relss 4427 | . . . 4 | |
4 | 1, 2, 3 | mp2 16 | . . 3 |
5 | 4 | a1i 9 | . 2 |
6 | simpr 103 | . . . . 5 | |
7 | brinxp2 4407 | . . . . 5 | |
8 | 6, 7 | sylib 127 | . . . 4 |
9 | 8 | simp2d 917 | . . 3 |
10 | 8 | simp1d 916 | . . 3 |
11 | erinxp.r | . . . . 5 | |
12 | 11 | adantr 261 | . . . 4 |
13 | 8 | simp3d 918 | . . . 4 |
14 | 12, 13 | ersym 6118 | . . 3 |
15 | brinxp2 4407 | . . 3 | |
16 | 9, 10, 14, 15 | syl3anbrc 1088 | . 2 |
17 | 10 | adantrr 448 | . . 3 |
18 | simprr 484 | . . . . 5 | |
19 | brinxp2 4407 | . . . . 5 | |
20 | 18, 19 | sylib 127 | . . . 4 |
21 | 20 | simp2d 917 | . . 3 |
22 | 11 | adantr 261 | . . . 4 |
23 | 13 | adantrr 448 | . . . 4 |
24 | 20 | simp3d 918 | . . . 4 |
25 | 22, 23, 24 | ertrd 6122 | . . 3 |
26 | brinxp2 4407 | . . 3 | |
27 | 17, 21, 25, 26 | syl3anbrc 1088 | . 2 |
28 | 11 | adantr 261 | . . . . . 6 |
29 | erinxp.a | . . . . . . 7 | |
30 | 29 | sselda 2945 | . . . . . 6 |
31 | 28, 30 | erref 6126 | . . . . 5 |
32 | 31 | ex 108 | . . . 4 |
33 | 32 | pm4.71rd 374 | . . 3 |
34 | brin 3811 | . . . 4 | |
35 | brxp 4375 | . . . . . 6 | |
36 | anidm 376 | . . . . . 6 | |
37 | 35, 36 | bitri 173 | . . . . 5 |
38 | 37 | anbi2i 430 | . . . 4 |
39 | 34, 38 | bitri 173 | . . 3 |
40 | 33, 39 | syl6bbr 187 | . 2 |
41 | 5, 16, 27, 40 | iserd 6132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wcel 1393 cin 2916 wss 2917 class class class wbr 3764 cxp 4343 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-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: (None) |
Copyright terms: Public domain | W3C validator |