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Theorem ecopover 7011
 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.)
Hypotheses
Ref Expression
ecopopr.1
ecopopr.com
ecopopr.cl
ecopopr.ass
ecopopr.can
Assertion
Ref Expression
ecopover
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)

Proof of Theorem ecopover
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . 5
21relopabi 5003 . . . 4
32a1i 11 . . 3
4 ecopopr.com . . . . 5
51, 4ecopovsym 7009 . . . 4
7 ecopopr.cl . . . . 5
8 ecopopr.ass . . . . 5
9 ecopopr.can . . . . 5
101, 4, 7, 8, 9ecopovtrn 7010 . . . 4
12 vex 2961 . . . . . . . . . . 11
13 vex 2961 . . . . . . . . . . 11
1412, 13, 4caovcom 6247 . . . . . . . . . 10
151ecopoveq 7008 . . . . . . . . . 10
1614, 15mpbiri 226 . . . . . . . . 9
1716anidms 628 . . . . . . . 8
1817rgen2a 2774 . . . . . . 7
19 breq12 4220 . . . . . . . . 9
2019anidms 628 . . . . . . . 8
2120ralxp 5019 . . . . . . 7
2218, 21mpbir 202 . . . . . 6
2322rspec 2772 . . . . 5
2423a1i 11 . . . 4
25 opabssxp 4953 . . . . . . 7
261, 25eqsstri 3380 . . . . . 6
2726ssbri 4257 . . . . 5
28 brxp 4912 . . . . . 6
2928simplbi 448 . . . . 5
3027, 29syl 16 . . . 4
3124, 30impbid1 196 . . 3
323, 6, 11, 31iserd 6934 . 2
3332trud 1333 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wtru 1326  wex 1551   wceq 1653   wcel 1726  wral 2707  cop 3819   class class class wbr 4215  copab 4268   cxp 4879   wrel 4886  (class class class)co 6084   wer 6905 This theorem is referenced by:  enqer  8803  enrer  8948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fv 5465  df-ov 6087  df-er 6908
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