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Theorem swopo 4505
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 20 . . . . 5  |-  ( x  e.  A  ->  x  e.  A )
21ancli 535 . . . 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 2790 . . . 4  |-  ( ph  ->  A. y  e.  A  A. z  e.  A  ( y R z  ->  -.  z R
y ) )
5 breq1 4207 . . . . . 6  |-  ( y  =  x  ->  (
y R z  <->  x R
z ) )
6 breq2 4208 . . . . . . 7  |-  ( y  =  x  ->  (
z R y  <->  z R x ) )
76notbid 286 . . . . . 6  |-  ( y  =  x  ->  ( -.  z R y  <->  -.  z R x ) )
85, 7imbi12d 312 . . . . 5  |-  ( y  =  x  ->  (
( y R z  ->  -.  z R
y )  <->  ( x R z  ->  -.  z R x ) ) )
9 breq2 4208 . . . . . 6  |-  ( z  =  x  ->  (
x R z  <->  x R x ) )
10 breq1 4207 . . . . . . 7  |-  ( z  =  x  ->  (
z R x  <->  x R x ) )
1110notbid 286 . . . . . 6  |-  ( z  =  x  ->  ( -.  z R x  <->  -.  x R x ) )
129, 11imbi12d 312 . . . . 5  |-  ( z  =  x  ->  (
( x R z  ->  -.  z R x )  <->  ( x R x  ->  -.  x R x ) ) )
138, 12rspc2va 3051 . . . 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 465 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x R x  ->  -.  x R x ) )
1514pm2.01d 163 . 2  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
1633adantr1 1116 . . 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 419 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
x R z  \/  z R y ) )
1918orcomd 378 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
z R y  \/  x R z ) )
2019ord 367 . . . 4  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  ( -.  z R y  ->  x R z ) )
2120expimpd 587 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  -.  z R y )  ->  x R z ) )
2216, 21sylan2d 469 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
2315, 22ispod 4503 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2697   class class class wbr 4204    Po wpo 4493
This theorem is referenced by:  swoer  6925  swoso  6928
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  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-br 4205  df-po 4495
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