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Mirrors > Home > ILE Home > Th. List > ecopover | Unicode version |
Description: Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopopr.com | |
ecopopr.cl | |
ecopopr.ass | |
ecopopr.can |
Ref | Expression |
---|---|
ecopover |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . 5 | |
2 | 1 | relopabi 4463 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | ecopopr.com | . . . . 5 | |
5 | 1, 4 | ecopovsym 6202 | . . . 4 |
6 | 5 | adantl 262 | . . 3 |
7 | ecopopr.cl | . . . . 5 | |
8 | ecopopr.ass | . . . . 5 | |
9 | ecopopr.can | . . . . 5 | |
10 | 1, 4, 7, 8, 9 | ecopovtrn 6203 | . . . 4 |
11 | 10 | adantl 262 | . . 3 |
12 | vex 2560 | . . . . . . . . . . 11 | |
13 | vex 2560 | . . . . . . . . . . 11 | |
14 | 12, 13, 4 | caovcom 5658 | . . . . . . . . . 10 |
15 | 1 | ecopoveq 6201 | . . . . . . . . . 10 |
16 | 14, 15 | mpbiri 157 | . . . . . . . . 9 |
17 | 16 | anidms 377 | . . . . . . . 8 |
18 | 17 | rgen2a 2375 | . . . . . . 7 |
19 | breq12 3769 | . . . . . . . . 9 | |
20 | 19 | anidms 377 | . . . . . . . 8 |
21 | 20 | ralxp 4479 | . . . . . . 7 |
22 | 18, 21 | mpbir 134 | . . . . . 6 |
23 | 22 | rspec 2373 | . . . . 5 |
24 | 23 | a1i 9 | . . . 4 |
25 | opabssxp 4414 | . . . . . . 7 | |
26 | 1, 25 | eqsstri 2975 | . . . . . 6 |
27 | 26 | ssbri 3806 | . . . . 5 |
28 | brxp 4375 | . . . . . 6 | |
29 | 28 | simplbi 259 | . . . . 5 |
30 | 27, 29 | syl 14 | . . . 4 |
31 | 24, 30 | impbid1 130 | . . 3 |
32 | 3, 6, 11, 31 | iserd 6132 | . 2 |
33 | 32 | trud 1252 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wtru 1244 wex 1381 wcel 1393 wral 2306 cop 3378 class class class wbr 3764 copab 3817 cxp 4343 wrel 4350 (class class class)co 5512 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-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fv 4910 df-ov 5515 df-er 6106 |
This theorem is referenced by: (None) |
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