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Theorem swoer 6320
Description: Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
swoer.1  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
swoer.2  |-  ( (
ph  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y  .<  z  ->  -.  z  .<  y
) )
swoer.3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( x  .<  y  ->  ( x  .<  z  \/  z  .<  y ) ) )
Assertion
Ref Expression
swoer  |-  ( ph  ->  R  Er  X )
Distinct variable groups:    x, y, z, 
.<   
ph, x, y, z   
x, X, y, z
Allowed substitution hints:    R( x, y, z)

Proof of Theorem swoer
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 swoer.1 . . . . 5  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
2 difss 3126 . . . . 5  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
31, 2eqsstri 3056 . . . 4  |-  R  C_  ( X  X.  X
)
4 relxp 4547 . . . 4  |-  Rel  ( X  X.  X )
5 relss 4525 . . . 4  |-  ( R 
C_  ( X  X.  X )  ->  ( Rel  ( X  X.  X
)  ->  Rel  R ) )
63, 4, 5mp2 16 . . 3  |-  Rel  R
76a1i 9 . 2  |-  ( ph  ->  Rel  R )
8 simpr 108 . . 3  |-  ( (
ph  /\  u R
v )  ->  u R v )
9 orcom 682 . . . . . 6  |-  ( ( u  .<  v  \/  v  .<  u )  <->  ( v  .<  u  \/  u  .<  v ) )
109a1i 9 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  (
( u  .<  v  \/  v  .<  u )  <-> 
( v  .<  u  \/  u  .<  v ) ) )
1110notbid 627 . . . 4  |-  ( (
ph  /\  u R
v )  ->  ( -.  ( u  .<  v  \/  v  .<  u )  <->  -.  ( v  .<  u  \/  u  .<  v ) ) )
123ssbri 3887 . . . . . . 7  |-  ( u R v  ->  u
( X  X.  X
) v )
1312adantl 271 . . . . . 6  |-  ( (
ph  /\  u R
v )  ->  u
( X  X.  X
) v )
14 brxp 4468 . . . . . 6  |-  ( u ( X  X.  X
) v  <->  ( u  e.  X  /\  v  e.  X ) )
1513, 14sylib 120 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  (
u  e.  X  /\  v  e.  X )
)
161brdifun 6319 . . . . 5  |-  ( ( u  e.  X  /\  v  e.  X )  ->  ( u R v  <->  -.  ( u  .<  v  \/  v  .<  u ) ) )
1715, 16syl 14 . . . 4  |-  ( (
ph  /\  u R
v )  ->  (
u R v  <->  -.  (
u  .<  v  \/  v  .<  u ) ) )
1815simprd 112 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  v  e.  X )
1915simpld 110 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  u  e.  X )
201brdifun 6319 . . . . 5  |-  ( ( v  e.  X  /\  u  e.  X )  ->  ( v R u  <->  -.  ( v  .<  u  \/  u  .<  v ) ) )
2118, 19, 20syl2anc 403 . . . 4  |-  ( (
ph  /\  u R
v )  ->  (
v R u  <->  -.  (
v  .<  u  \/  u  .<  v ) ) )
2211, 17, 213bitr4d 218 . . 3  |-  ( (
ph  /\  u R
v )  ->  (
u R v  <->  v R u ) )
238, 22mpbid 145 . 2  |-  ( (
ph  /\  u R
v )  ->  v R u )
24 simprl 498 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u R v )
2512ad2antrl 474 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u ( X  X.  X ) v )
2614simplbi 268 . . . . . . 7  |-  ( u ( X  X.  X
) v  ->  u  e.  X )
2725, 26syl 14 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u  e.  X
)
2814simprbi 269 . . . . . . 7  |-  ( u ( X  X.  X
) v  ->  v  e.  X )
2925, 28syl 14 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  v  e.  X
)
3027, 29, 16syl2anc 403 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u R v  <->  -.  ( u  .<  v  \/  v  .<  u ) ) )
3124, 30mpbid 145 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( u  .<  v  \/  v  .<  u ) )
32 simprr 499 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  v R w )
333brel 4490 . . . . . . . 8  |-  ( v R w  ->  (
v  e.  X  /\  w  e.  X )
)
3433simprd 112 . . . . . . 7  |-  ( v R w  ->  w  e.  X )
3532, 34syl 14 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  w  e.  X
)
361brdifun 6319 . . . . . 6  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v R w  <->  -.  ( v  .<  w  \/  w  .<  v ) ) )
3729, 35, 36syl2anc 403 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( v R w  <->  -.  ( v  .<  w  \/  w  .<  v ) ) )
3832, 37mpbid 145 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( v  .<  w  \/  w  .<  v ) )
39 simpl 107 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ph )
40 swoer.3 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( x  .<  y  ->  ( x  .<  z  \/  z  .<  y ) ) )
4140swopolem 4132 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  X  /\  w  e.  X  /\  v  e.  X ) )  -> 
( u  .<  w  ->  ( u  .<  v  \/  v  .<  w ) ) )
4239, 27, 35, 29, 41syl13anc 1176 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u  .<  w  ->  ( u  .<  v  \/  v  .<  w
) ) )
4340swopolem 4132 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  X  /\  u  e.  X  /\  v  e.  X ) )  -> 
( w  .<  u  ->  ( w  .<  v  \/  v  .<  u ) ) )
4439, 35, 27, 29, 43syl13anc 1176 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( w  .<  u  ->  ( w  .<  v  \/  v  .<  u
) ) )
45 orcom 682 . . . . . . 7  |-  ( ( v  .<  u  \/  w  .<  v )  <->  ( w  .<  v  \/  v  .<  u ) )
4644, 45syl6ibr 160 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( w  .<  u  ->  ( v  .<  u  \/  w  .<  v ) ) )
4742, 46orim12d 735 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( ( u 
.<  w  \/  w  .<  u )  ->  (
( u  .<  v  \/  v  .<  w )  \/  ( v  .<  u  \/  w  .<  v ) ) ) )
48 or4 723 . . . . 5  |-  ( ( ( u  .<  v  \/  v  .<  w )  \/  ( v  .<  u  \/  w  .<  v ) )  <->  ( (
u  .<  v  \/  v  .<  u )  \/  (
v  .<  w  \/  w  .<  v ) ) )
4947, 48syl6ib 159 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( ( u 
.<  w  \/  w  .<  u )  ->  (
( u  .<  v  \/  v  .<  u )  \/  ( v  .<  w  \/  w  .<  v ) ) ) )
5031, 38, 49mtord 732 . . 3  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( u  .<  w  \/  w  .<  u ) )
511brdifun 6319 . . . 4  |-  ( ( u  e.  X  /\  w  e.  X )  ->  ( u R w  <->  -.  ( u  .<  w  \/  w  .<  u ) ) )
5227, 35, 51syl2anc 403 . . 3  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u R w  <->  -.  ( u  .<  w  \/  w  .<  u ) ) )
5350, 52mpbird 165 . 2  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u R w )
54 swoer.2 . . . . . . 7  |-  ( (
ph  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y  .<  z  ->  -.  z  .<  y
) )
5554, 40swopo 4133 . . . . . 6  |-  ( ph  ->  .<  Po  X )
56 poirr 4134 . . . . . 6  |-  ( ( 
.<  Po  X  /\  u  e.  X )  ->  -.  u  .<  u )
5755, 56sylan 277 . . . . 5  |-  ( (
ph  /\  u  e.  X )  ->  -.  u  .<  u )
58 pm1.2 708 . . . . 5  |-  ( ( u  .<  u  \/  u  .<  u )  ->  u  .<  u )
5957, 58nsyl 593 . . . 4  |-  ( (
ph  /\  u  e.  X )  ->  -.  ( u  .<  u  \/  u  .<  u )
)
60 simpr 108 . . . . 5  |-  ( (
ph  /\  u  e.  X )  ->  u  e.  X )
611brdifun 6319 . . . . 5  |-  ( ( u  e.  X  /\  u  e.  X )  ->  ( u R u  <->  -.  ( u  .<  u  \/  u  .<  u ) ) )
6260, 60, 61syl2anc 403 . . . 4  |-  ( (
ph  /\  u  e.  X )  ->  (
u R u  <->  -.  (
u  .<  u  \/  u  .<  u ) ) )
6359, 62mpbird 165 . . 3  |-  ( (
ph  /\  u  e.  X )  ->  u R u )
643ssbri 3887 . . . . 5  |-  ( u R u  ->  u
( X  X.  X
) u )
65 brxp 4468 . . . . . 6  |-  ( u ( X  X.  X
) u  <->  ( u  e.  X  /\  u  e.  X ) )
6665simplbi 268 . . . . 5  |-  ( u ( X  X.  X
) u  ->  u  e.  X )
6764, 66syl 14 . . . 4  |-  ( u R u  ->  u  e.  X )
6867adantl 271 . . 3  |-  ( (
ph  /\  u R u )  ->  u  e.  X )
6963, 68impbida 563 . 2  |-  ( ph  ->  ( u  e.  X  <->  u R u ) )
707, 23, 53, 69iserd 6318 1  |-  ( ph  ->  R  Er  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438    \ cdif 2996    u. cun 2997    C_ wss 2999   class class class wbr 3845    Po wpo 4121    X. cxp 4436   `'ccnv 4437   Rel wrel 4443    Er wer 6289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-po 4123  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-er 6292
This theorem is referenced by: (None)
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