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Theorem swopo 4650
Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
swopo.1  |-  ( (
ph  /\  ( y  e.  A  /\  z  e.  A ) )  -> 
( y R z  ->  -.  z R
y ) )
swopo.2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( x R y  ->  ( x R z  \/  z R y ) ) )
Assertion
Ref Expression
swopo  |-  ( ph  ->  R  Po  A )
Distinct variable groups:    x, y,
z, A    x, R, y, z    ph, x, y, z

Proof of Theorem swopo
StepHypRef Expression
1 id 22 . . . . 5  |-  ( x  e.  A  ->  x  e.  A )
21ancli 551 . . . 4  |-  ( x  e.  A  ->  (
x  e.  A  /\  x  e.  A )
)
3 swopo.1 . . . . 5  |-  ( (
ph  /\  ( y  e.  A  /\  z  e.  A ) )  -> 
( y R z  ->  -.  z R
y ) )
43ralrimivva 2807 . . . 4  |-  ( ph  ->  A. y  e.  A  A. z  e.  A  ( y R z  ->  -.  z R
y ) )
5 breq1 4294 . . . . . 6  |-  ( y  =  x  ->  (
y R z  <->  x R
z ) )
6 breq2 4295 . . . . . . 7  |-  ( y  =  x  ->  (
z R y  <->  z R x ) )
76notbid 294 . . . . . 6  |-  ( y  =  x  ->  ( -.  z R y  <->  -.  z R x ) )
85, 7imbi12d 320 . . . . 5  |-  ( y  =  x  ->  (
( y R z  ->  -.  z R
y )  <->  ( x R z  ->  -.  z R x ) ) )
9 breq2 4295 . . . . . 6  |-  ( z  =  x  ->  (
x R z  <->  x R x ) )
10 breq1 4294 . . . . . . 7  |-  ( z  =  x  ->  (
z R x  <->  x R x ) )
1110notbid 294 . . . . . 6  |-  ( z  =  x  ->  ( -.  z R x  <->  -.  x R x ) )
129, 11imbi12d 320 . . . . 5  |-  ( z  =  x  ->  (
( x R z  ->  -.  z R x )  <->  ( x R x  ->  -.  x R x ) ) )
138, 12rspc2va 3079 . . . 4  |-  ( ( ( x  e.  A  /\  x  e.  A
)  /\  A. y  e.  A  A. z  e.  A  ( y R z  ->  -.  z R y ) )  ->  ( x R x  ->  -.  x R x ) )
142, 4, 13syl2anr 478 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x R x  ->  -.  x R x ) )
1514pm2.01d 169 . 2  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
1633adantr1 1147 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( y R z  ->  -.  z R
y ) )
17 swopo.2 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( x R y  ->  ( x R z  \/  z R y ) ) )
1817imp 429 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
x R z  \/  z R y ) )
1918orcomd 388 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
z R y  \/  x R z ) )
2019ord 377 . . . 4  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  ( -.  z R y  ->  x R z ) )
2120expimpd 603 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  -.  z R y )  ->  x R z ) )
2216, 21sylan2d 482 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
2315, 22ispod 4648 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    e. wcel 1756   A.wral 2714   class class class wbr 4291    Po wpo 4638
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
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-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2719  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-po 4640
This theorem is referenced by:  swoer  7128  swoso  7131
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