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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 4481 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | ecopopr.com | . . . . 5 | |
5 | 1, 4 | ecopovsym 6225 | . . . 4 |
6 | 5 | adantl 271 | . . 3 |
7 | ecopopr.cl | . . . . 5 | |
8 | ecopopr.ass | . . . . 5 | |
9 | ecopopr.can | . . . . 5 | |
10 | 1, 4, 7, 8, 9 | ecopovtrn 6226 | . . . 4 |
11 | 10 | adantl 271 | . . 3 |
12 | vex 2604 | . . . . . . . . . . 11 | |
13 | vex 2604 | . . . . . . . . . . 11 | |
14 | 12, 13, 4 | caovcom 5678 | . . . . . . . . . 10 |
15 | 1 | ecopoveq 6224 | . . . . . . . . . 10 |
16 | 14, 15 | mpbiri 166 | . . . . . . . . 9 |
17 | 16 | anidms 389 | . . . . . . . 8 |
18 | 17 | rgen2a 2417 | . . . . . . 7 |
19 | breq12 3790 | . . . . . . . . 9 | |
20 | 19 | anidms 389 | . . . . . . . 8 |
21 | 20 | ralxp 4497 | . . . . . . 7 |
22 | 18, 21 | mpbir 144 | . . . . . 6 |
23 | 22 | rspec 2415 | . . . . 5 |
24 | 23 | a1i 9 | . . . 4 |
25 | opabssxp 4432 | . . . . . . 7 | |
26 | 1, 25 | eqsstri 3029 | . . . . . 6 |
27 | 26 | ssbri 3827 | . . . . 5 |
28 | brxp 4393 | . . . . . 6 | |
29 | 28 | simplbi 268 | . . . . 5 |
30 | 27, 29 | syl 14 | . . . 4 |
31 | 24, 30 | impbid1 140 | . . 3 |
32 | 3, 6, 11, 31 | iserd 6155 | . 2 |
33 | 32 | trud 1293 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wtru 1285 wex 1421 wcel 1433 wral 2348 cop 3401 class class class wbr 3785 copab 3838 cxp 4361 wrel 4368 (class class class)co 5532 wer 6126 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fv 4930 df-ov 5535 df-er 6129 |
This theorem is referenced by: (None) |
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