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Theorem cotr 5239
 Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cotr
Distinct variable group:   ,,,

Proof of Theorem cotr
StepHypRef Expression
1 df-co 4880 . . . 4
21relopabi 4993 . . 3
3 ssrel 4957 . . 3
42, 3ax-mp 8 . 2
5 vex 2952 . . . . . . . 8
6 vex 2952 . . . . . . . 8
75, 6opelco 5037 . . . . . . 7
8 df-br 4206 . . . . . . . 8
98bicomi 194 . . . . . . 7
107, 9imbi12i 317 . . . . . 6
11 19.23v 1914 . . . . . 6
1210, 11bitr4i 244 . . . . 5
1312albii 1575 . . . 4
14 alcom 1752 . . . 4
1513, 14bitri 241 . . 3
1615albii 1575 . 2
174, 16bitri 241 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549  wex 1550   wcel 1725   wss 3313  cop 3810   class class class wbr 4205   ccom 4875   wrel 4876 This theorem is referenced by:  xpidtr  5249  trin2  5250  dfer2  6899  pslem  14631  letsr  14665  dirtr  14674  filnetlem3  26401 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-rel 4878  df-co 4880
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