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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 4724 | . . . 4 |
3 | 2 | a1i 9 | . . 3 |
4 | ecopopr.com | . . . . 5 | |
5 | 1, 4 | ecopovsym 6588 | . . . 4 |
6 | 5 | adantl 275 | . . 3 |
7 | ecopopr.cl | . . . . 5 | |
8 | ecopopr.ass | . . . . 5 | |
9 | ecopopr.can | . . . . 5 | |
10 | 1, 4, 7, 8, 9 | ecopovtrn 6589 | . . . 4 |
11 | 10 | adantl 275 | . . 3 |
12 | vex 2724 | . . . . . . . . . . 11 | |
13 | vex 2724 | . . . . . . . . . . 11 | |
14 | 12, 13, 4 | caovcom 5990 | . . . . . . . . . 10 |
15 | 1 | ecopoveq 6587 | . . . . . . . . . 10 |
16 | 14, 15 | mpbiri 167 | . . . . . . . . 9 |
17 | 16 | anidms 395 | . . . . . . . 8 |
18 | 17 | rgen2a 2518 | . . . . . . 7 |
19 | breq12 3981 | . . . . . . . . 9 | |
20 | 19 | anidms 395 | . . . . . . . 8 |
21 | 20 | ralxp 4741 | . . . . . . 7 |
22 | 18, 21 | mpbir 145 | . . . . . 6 |
23 | 22 | rspec 2516 | . . . . 5 |
24 | 23 | a1i 9 | . . . 4 |
25 | opabssxp 4672 | . . . . . . 7 | |
26 | 1, 25 | eqsstri 3169 | . . . . . 6 |
27 | 26 | ssbri 4020 | . . . . 5 |
28 | brxp 4629 | . . . . . 6 | |
29 | 28 | simplbi 272 | . . . . 5 |
30 | 27, 29 | syl 14 | . . . 4 |
31 | 24, 30 | impbid1 141 | . . 3 |
32 | 3, 6, 11, 31 | iserd 6518 | . 2 |
33 | 32 | mptru 1351 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wtru 1343 wex 1479 wcel 2135 wral 2442 cop 3573 class class class wbr 3976 copab 4036 cxp 4596 wrel 4603 (class class class)co 5836 wer 6489 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fv 5190 df-ov 5839 df-er 6492 |
This theorem is referenced by: (None) |
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