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Theorem preotr2 25235
 Description: A preset is transitive. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
preotr2 PresetRel

Proof of Theorem preotr2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preotr1 25234 . . . 4 PresetRel
21adantr 451 . . 3 PresetRel
3 cotr 5055 . . 3
42, 3sylib 188 . 2 PresetRel
5 simpr 447 . 2 PresetRel
6 preorel 25225 . . 3 PresetRel
7 brrelex 4727 . . . . 5
87adantrr 697 . . . 4
9 brrelex2 4728 . . . . 5
109adantrr 697 . . . 4
11 brrelex2 4728 . . . . 5
1211adantrl 696 . . . 4
13 simp1 955 . . . . . . . 8
14 simp2 956 . . . . . . . 8
1513, 14breq12d 4036 . . . . . . 7
16 simp3 957 . . . . . . . 8
1714, 16breq12d 4036 . . . . . . 7
1815, 17anbi12d 691 . . . . . 6
1913, 16breq12d 4036 . . . . . 6
2018, 19imbi12d 311 . . . . 5
2120spc3gv 2873 . . . 4
228, 10, 12, 21syl3anc 1182 . . 3
236, 22sylan 457 . 2 PresetRel
244, 5, 23mp2d 41 1 PresetRel
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934  wal 1527   wceq 1623   wcel 1684  cvv 2788   wss 3152   class class class wbr 4023   ccom 4693   wrel 4694  PresetRelcpresetrel 25215 This theorem is referenced by:  prltub  25260 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-co 4698  df-res 4701  df-prs 25223
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