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Theorem pslem 16388
Description: Lemma for psref 16390 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
pslem  |-  ( R  e.  PosetRel  ->  ( ( ( A R B  /\  B R C )  ->  A R C )  /\  ( A  e.  U. U. R  ->  A R A )  /\  ( ( A R B  /\  B R A )  ->  A  =  B )
) )

Proof of Theorem pslem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrel 16385 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
2 brrelex12 4827 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
31, 2sylan 473 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
4 brrelex2 4829 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
51, 4sylan 473 . . . . 5  |-  ( ( R  e.  PosetRel  /\  B R C )  ->  C  e.  _V )
63, 5anim12dan 845 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  /\  C  e. 
_V ) )
7 pstr2 16387 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
8 cotr 5167 . . . . . 6  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
97, 8sylib 199 . . . . 5  |-  ( R  e.  PosetRel  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
109adantr 466 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
11 simpr 462 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
12 breq12 4364 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x R y  <-> 
A R B ) )
13123adant3 1025 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
14 breq12 4364 . . . . . . . . 9  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
15143adant1 1023 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1613, 15anbi12d 715 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
17 breq12 4364 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
18173adant2 1024 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
1916, 18imbi12d 321 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( ( x R y  /\  y R z )  ->  x R z )  <->  ( ( A R B  /\  B R C )  ->  A R C ) ) )
2019spc3gv 3107 . . . . 5  |-  ( ( 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 ) ) )
21203expa 1205 . . . 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 ) ) )
226, 10, 11, 21syl3c 63 . . 3  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )
2322ex 435 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
24 psref2 16386 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
25 asymref2 5172 . . . 4  |-  ( ( R  i^i  `' R
)  =  (  _I  |`  U. U. R )  <-> 
( A. x  e. 
U. U. R x R x  /\  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y )
) )
2625simplbi 461 . . 3  |-  ( ( R  i^i  `' R
)  =  (  _I  |`  U. U. R )  ->  A. x  e.  U. U. R x R x )
27 breq12 4364 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
2827anidms 649 . . . 4  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
2928rspccv 3115 . . 3  |-  ( A. x  e.  U. U. R x R x  ->  ( A  e.  U. U. R  ->  A R A ) )
3024, 26, 293syl 18 . 2  |-  ( R  e.  PosetRel  ->  ( A  e. 
U. U. R  ->  A R A ) )
313adantrr 721 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
3225simprbi 465 . . . . . 6  |-  ( ( R  i^i  `' R
)  =  (  _I  |`  U. U. R )  ->  A. x A. y
( ( x R y  /\  y R x )  ->  x  =  y ) )
3324, 32syl 17 . . . . 5  |-  ( R  e.  PosetRel  ->  A. x A. y
( ( x R y  /\  y R x )  ->  x  =  y ) )
3433adantr 466 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  ->  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y ) )
35 simpr 462 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  -> 
( A R B  /\  B R A ) )
36 breq12 4364 . . . . . . . 8  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y R x  <-> 
B R A ) )
3736ancoms 454 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y R x  <-> 
B R A ) )
3812, 37anbi12d 715 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x R y  /\  y R x )  <->  ( A R B  /\  B R A ) ) )
39 eqeq12 2435 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  y  <-> 
A  =  B ) )
4038, 39imbi12d 321 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x R y  /\  y R x )  ->  x  =  y )  <->  ( ( A R B  /\  B R A )  ->  A  =  B ) ) )
4140spc2gv 3105 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y )  ->  ( ( A R B  /\  B R A )  ->  A  =  B ) ) )
4231, 34, 35, 41syl3c 63 . . 3  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  ->  A  =  B )
4342ex 435 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R A )  ->  A  =  B ) )
4423, 30, 433jca 1185 1  |-  ( R  e.  PosetRel  ->  ( ( ( A R B  /\  B R C )  ->  A R C )  /\  ( A  e.  U. U. R  ->  A R A )  /\  ( ( A R B  /\  B R A )  ->  A  =  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1872   A.wral 2708   _Vcvv 3016    i^i cin 3371    C_ wss 3372   U.cuni 4155   class class class wbr 4359    _I cid 4699   `'ccnv 4788    |` cres 4791    o. ccom 4793   Rel wrel 4794   PosetRelcps 16380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pr 4596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-res 4801  df-ps 16382
This theorem is referenced by:  psdmrn  16389  psref  16390  psasym  16392  pstr  16393
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