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Theorem pslem 15374
Description: Lemma for psref 15376 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 15371 . . . . . 6  |-  ( R  e.  PosetRel  ->  Rel  R )
2 brrelex12 4874 . . . . . 6  |-  ( ( Rel  R  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
31, 2sylan 471 . . . . 5  |-  ( ( R  e.  PosetRel  /\  A R B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
4 brrelex2 4876 . . . . . 6  |-  ( ( Rel  R  /\  B R C )  ->  C  e.  _V )
51, 4sylan 471 . . . . 5  |-  ( ( R  e.  PosetRel  /\  B R C )  ->  C  e.  _V )
63, 5anim12dan 833 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  -> 
( ( A  e. 
_V  /\  B  e.  _V )  /\  C  e. 
_V ) )
7 pstr2 15373 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( R  o.  R )  C_  R
)
8 cotr 5208 . . . . . 6  |-  ( ( R  o.  R ) 
C_  R  <->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R
z ) )
97, 8sylib 196 . . . . 5  |-  ( R  e.  PosetRel  ->  A. x A. y A. z ( ( x R y  /\  y R z )  ->  x R z ) )
109adantr 465 . . . 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 461 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  -> 
( A R B  /\  B R C ) )
12 breq12 4295 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x R y  <-> 
A R B ) )
13123adant3 1008 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R y  <-> 
A R B ) )
14 breq12 4295 . . . . . . . . 9  |-  ( ( y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
15143adant1 1006 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( y R z  <-> 
B R C ) )
1613, 15anbi12d 710 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ( x R y  /\  y R z )  <->  ( A R B  /\  B R C ) ) )
17 breq12 4295 . . . . . . . 8  |-  ( ( x  =  A  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
18173adant2 1007 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( x R z  <-> 
A R C ) )
1916, 18imbi12d 320 . . . . . 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 3060 . . . . 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 1187 . . . 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 61 . . 3  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )
2322ex 434 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R C )  ->  A R C ) )
24 psref2 15372 . . 3  |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |`  U. U. R
) )
25 asymref2 5213 . . . 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 460 . . 3  |-  ( ( R  i^i  `' R
)  =  (  _I  |`  U. U. R )  ->  A. x  e.  U. U. R x R x )
27 breq12 4295 . . . . 5  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
2827anidms 645 . . . 4  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
2928rspccv 3068 . . 3  |-  ( A. x  e.  U. U. R x R x  ->  ( A  e.  U. U. R  ->  A R A ) )
3024, 26, 293syl 20 . 2  |-  ( R  e.  PosetRel  ->  ( A  e. 
U. U. R  ->  A R A ) )
313adantrr 716 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  -> 
( A  e.  _V  /\  B  e.  _V )
)
3225simprbi 464 . . . . . 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 465 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  ->  A. x A. y ( ( x R y  /\  y R x )  ->  x  =  y ) )
35 simpr 461 . . . 4  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  -> 
( A R B  /\  B R A ) )
36 breq12 4295 . . . . . . . 8  |-  ( ( y  =  B  /\  x  =  A )  ->  ( y R x  <-> 
B R A ) )
3736ancoms 453 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y R x  <-> 
B R A ) )
3812, 37anbi12d 710 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x R y  /\  y R x )  <->  ( A R B  /\  B R A ) ) )
39 eqeq12 2453 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  y  <-> 
A  =  B ) )
4038, 39imbi12d 320 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( x R y  /\  y R x )  ->  x  =  y )  <->  ( ( A R B  /\  B R A )  ->  A  =  B ) ) )
4140spc2gv 3058 . . . 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 61 . . 3  |-  ( ( R  e.  PosetRel  /\  ( A R B  /\  B R A ) )  ->  A  =  B )
4342ex 434 . 2  |-  ( R  e.  PosetRel  ->  ( ( A R B  /\  B R A )  ->  A  =  B ) )
4423, 30, 433jca 1168 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 184    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2713   _Vcvv 2970    i^i cin 3325    C_ wss 3326   U.cuni 4089   class class class wbr 4290    _I cid 4629   `'ccnv 4837    |` cres 4840    o. ccom 4842   Rel wrel 4843   PosetRelcps 15366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-res 4850  df-ps 15368
This theorem is referenced by:  psdmrn  15375  psref  15376  psasym  15378  pstr  15379
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