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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  |-  ( ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )

Proof of Theorem preotr2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preotr1 25234 . . . 4  |-  ( R  e. PresetRel  ->  ( R  o.  R )  C_  R
)
21adantr 451 . . 3  |-  ( ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  -> 
( R  o.  R
)  C_  R )
3 cotr 5055 . . 3  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
42, 3sylib 188 . 2  |-  ( ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
5 simpr 447 . 2  |-  ( ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
6 preorel 25225 . . 3  |-  ( R  e. PresetRel  ->  Rel  R )
7 brrelex 4727 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  A  e.  _V )
87adantrr 697 . . . 4  |-  ( ( Rel  R  /\  ( A R B  /\  B R C ) )  ->  A  e.  _V )
9 brrelex2 4728 . . . . 5  |-  ( ( Rel  R  /\  A R B )  ->  B  e.  _V )
109adantrr 697 . . . 4  |-  ( ( Rel  R  /\  ( A R B  /\  B R C ) )  ->  B  e.  _V )
11 brrelex2 4728 . . . . 5  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
1211adantrl 696 . . . 4  |-  ( ( Rel  R  /\  ( A R B  /\  B R C ) )  ->  C  e.  _V )
13 simp1 955 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  x  =  A )
14 simp2 956 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  y  =  B )
1513, 14breq12d 4036 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
16 simp3 957 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  z  =  C )
1714, 16breq12d 4036 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1815, 17anbi12d 691 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
1913, 16breq12d 4036 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
2018, 19imbi12d 311 . . . . 5  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x R y  /\  y R z )  ->  x R z )  <->  ( ( A R B  /\  B R C )  ->  A R C ) ) )
2120spc3gv 2873 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z )  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) )
228, 10, 12, 21syl3anc 1182 . . 3  |-  ( ( Rel  R  /\  ( A R B  /\  B R C ) )  -> 
( A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z )  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) )
236, 22sylan 457 . 2  |-  ( ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  -> 
( A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z )  -> 
( ( A R B  /\  B R C )  ->  A R C ) ) )
244, 5, 23mp2d 41 1  |-  ( ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    o. ccom 4693   Rel 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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