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Theorem swopo 4340
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 19 . . . . 5  |-  ( x  e.  A  ->  x  e.  A )
21ancli 534 . . . 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 2648 . . . 4  |-  ( ph  ->  A. y  e.  A  A. z  e.  A  ( y R z  ->  -.  z R
y ) )
5 breq1 4042 . . . . . 6  |-  ( y  =  x  ->  (
y R z  <->  x R
z ) )
6 breq2 4043 . . . . . . 7  |-  ( y  =  x  ->  (
z R y  <->  z R x ) )
76notbid 285 . . . . . 6  |-  ( y  =  x  ->  ( -.  z R y  <->  -.  z R x ) )
85, 7imbi12d 311 . . . . 5  |-  ( y  =  x  ->  (
( y R z  ->  -.  z R
y )  <->  ( x R z  ->  -.  z R x ) ) )
9 breq2 4043 . . . . . 6  |-  ( z  =  x  ->  (
x R z  <->  x R x ) )
10 breq1 4042 . . . . . . 7  |-  ( z  =  x  ->  (
z R x  <->  x R x ) )
1110notbid 285 . . . . . 6  |-  ( z  =  x  ->  ( -.  z R x  <->  -.  x R x ) )
129, 11imbi12d 311 . . . . 5  |-  ( z  =  x  ->  (
( x R z  ->  -.  z R x )  <->  ( x R x  ->  -.  x R x ) ) )
138, 12rspc2va 2904 . . . 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 464 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x R x  ->  -.  x R x ) )
1514pm2.01d 161 . 2  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
1633adantr1 1114 . . 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 418 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
x R z  \/  z R y ) )
1918orcomd 377 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
z R y  \/  x R z ) )
2019ord 366 . . . 4  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  ( -.  z R y  ->  x R z ) )
2120expimpd 586 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  -.  z R y )  ->  x R z ) )
2216, 21sylan2d 468 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
2315, 22ispod 4338 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    Po wpo 4328
This theorem is referenced by:  swoer  6704  swoso  6707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-po 4330
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