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Theorem swopo 4223
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 321 . . . 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 2512 . . . 4  |-  ( ph  ->  A. y  e.  A  A. z  e.  A  ( y R z  ->  -.  z R
y ) )
5 breq1 3927 . . . . . 6  |-  ( y  =  x  ->  (
y R z  <->  x R
z ) )
6 breq2 3928 . . . . . . 7  |-  ( y  =  x  ->  (
z R y  <->  z R x ) )
76notbid 656 . . . . . 6  |-  ( y  =  x  ->  ( -.  z R y  <->  -.  z R x ) )
85, 7imbi12d 233 . . . . 5  |-  ( y  =  x  ->  (
( y R z  ->  -.  z R
y )  <->  ( x R z  ->  -.  z R x ) ) )
9 breq2 3928 . . . . . 6  |-  ( z  =  x  ->  (
x R z  <->  x R x ) )
10 breq1 3927 . . . . . . 7  |-  ( z  =  x  ->  (
z R x  <->  x R x ) )
1110notbid 656 . . . . . 6  |-  ( z  =  x  ->  ( -.  z R x  <->  -.  x R x ) )
129, 11imbi12d 233 . . . . 5  |-  ( z  =  x  ->  (
( x R z  ->  -.  z R x )  <->  ( x R x  ->  -.  x R x ) ) )
138, 12rspc2va 2798 . . . 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 288 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  (
x R x  ->  -.  x R x ) )
1514pm2.01d 607 . 2  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
1633adantr1 1140 . . 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 123 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
x R z  \/  z R y ) )
1918orcomd 718 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  (
z R y  \/  x R z ) )
2019ord 713 . . . 4  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  A  /\  z  e.  A )
)  /\  x R
y )  ->  ( -.  z R y  ->  x R z ) )
2120expimpd 360 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  -.  z R y )  ->  x R z ) )
2216, 21sylan2d 292 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
2315, 22ispod 4221 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697    /\ w3a 962    e. wcel 1480   A.wral 2414   class class class wbr 3924    Po wpo 4211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-po 4213
This theorem is referenced by:  swoer  6450
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