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Theorem pslem 14630
Description: Lemma for psref 14632 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 14627 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
2 brrelex12 4907 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
31, 2sylan 458 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
4 brrelex2 4909 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
51, 4sylan 458 . . . . 5  |-  ( ( R  e.  PosetRel  /\  B R C )  ->  C  e.  _V )
63, 5anim12dan 811 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  /\  C  e. 
_V ) )
7 pstr2 14629 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
8 cotr 5238 . . . . . 6  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
97, 8sylib 189 . . . . 5  |-  ( R  e.  PosetRel  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
109adantr 452 . . . 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 448 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
12 breq12 4209 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x R y  <-> 
A R B ) )
13123adant3 977 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
14 breq12 4209 . . . . . . . . 9  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
15143adant1 975 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1613, 15anbi12d 692 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
17 breq12 4209 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
18173adant2 976 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
1916, 18imbi12d 312 . . . . . 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 3033 . . . . 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 1153 . . . 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 59 . . 3  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )
2322ex 424 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
24 psref2 14628 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
25 asymref2 5243 . . . 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 447 . . 3  |-  ( ( R  i^i  `' R
)  =  (  _I  |`  U. U. R )  ->  A. x  e.  U. U. R x R x )
27 breq12 4209 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
2827anidms 627 . . . 4  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
2928rspccv 3041 . . 3  |-  ( A. x  e.  U. U. R x R x  ->  ( A  e.  U. U. R  ->  A R A ) )
3024, 26, 293syl 19 . 2  |-  ( R  e.  PosetRel  ->  ( A  e. 
U. U. R  ->  A R A ) )
313adantrr 698 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
3225simprbi 451 . . . . . 6  |-  ( ( R  i^i  `' R
)  =  (  _I  |`  U. U. R )  ->  A. x A. y
( ( x R y  /\  y R x )  ->  x  =  y ) )
3324, 32syl 16 . . . . 5  |-  ( R  e.  PosetRel  ->  A. x A. y
( ( x R y  /\  y R x )  ->  x  =  y ) )
3433adantr 452 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  ->  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y ) )
35 simpr 448 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  -> 
( A R B  /\  B R A ) )
36 breq12 4209 . . . . . . . 8  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y R x  <-> 
B R A ) )
3736ancoms 440 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y R x  <-> 
B R A ) )
3812, 37anbi12d 692 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x R y  /\  y R x )  <->  ( A R B  /\  B R A ) ) )
39 eqeq12 2447 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  y  <-> 
A  =  B ) )
4038, 39imbi12d 312 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x R y  /\  y R x )  ->  x  =  y )  <->  ( ( A R B  /\  B R A )  ->  A  =  B ) ) )
4140spc2gv 3031 . . . 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 59 . . 3  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  ->  A  =  B )
4342ex 424 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R A )  ->  A  =  B ) )
4423, 30, 433jca 1134 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   U.cuni 4007   class class class wbr 4204    _I cid 4485   `'ccnv 4869    |` cres 4872    o. ccom 4874   Rel wrel 4875   PosetRelcps 14616
This theorem is referenced by:  psdmrn  14631  psref  14632  psasym  14634  pstr  14635
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-res 4882  df-ps 14621
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