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Theorem swoer 7378
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 3572 . . . . 5  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
31, 2eqsstri 3474 . . . 4  |-  R  C_  ( X  X.  X
)
4 relxp 4933 . . . 4  |-  Rel  ( X  X.  X )
5 relss 4913 . . . 4  |-  ( R 
C_  ( X  X.  X )  ->  ( Rel  ( X  X.  X
)  ->  Rel  R ) )
63, 4, 5mp2 9 . . 3  |-  Rel  R
76a1i 11 . 2  |-  ( ph  ->  Rel  R )
8 simpr 461 . . 3  |-  ( (
ph  /\  u R
v )  ->  u R v )
9 orcom 387 . . . . . 6  |-  ( ( u  .<  v  \/  v  .<  u )  <->  ( v  .<  u  \/  u  .<  v ) )
109a1i 11 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  (
( u  .<  v  \/  v  .<  u )  <-> 
( v  .<  u  \/  u  .<  v ) ) )
1110notbid 294 . . . 4  |-  ( (
ph  /\  u R
v )  ->  ( -.  ( u  .<  v  \/  v  .<  u )  <->  -.  ( v  .<  u  \/  u  .<  v ) ) )
123ssbri 4439 . . . . . . 7  |-  ( u R v  ->  u
( X  X.  X
) v )
1312adantl 466 . . . . . 6  |-  ( (
ph  /\  u R
v )  ->  u
( X  X.  X
) v )
14 brxp 4856 . . . . . 6  |-  ( u ( X  X.  X
) v  <->  ( u  e.  X  /\  v  e.  X ) )
1513, 14sylib 198 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  (
u  e.  X  /\  v  e.  X )
)
161brdifun 7377 . . . . 5  |-  ( ( u  e.  X  /\  v  e.  X )  ->  ( u R v  <->  -.  ( u  .<  v  \/  v  .<  u ) ) )
1715, 16syl 17 . . . 4  |-  ( (
ph  /\  u R
v )  ->  (
u R v  <->  -.  (
u  .<  v  \/  v  .<  u ) ) )
1815simprd 463 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  v  e.  X )
1915simpld 459 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  u  e.  X )
201brdifun 7377 . . . . 5  |-  ( ( v  e.  X  /\  u  e.  X )  ->  ( v R u  <->  -.  ( v  .<  u  \/  u  .<  v ) ) )
2118, 19, 20syl2anc 661 . . . 4  |-  ( (
ph  /\  u R
v )  ->  (
v R u  <->  -.  (
v  .<  u  \/  u  .<  v ) ) )
2211, 17, 213bitr4d 287 . . 3  |-  ( (
ph  /\  u R
v )  ->  (
u R v  <->  v R u ) )
238, 22mpbid 212 . 2  |-  ( (
ph  /\  u R
v )  ->  v R u )
24 simprl 758 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u R v )
2512ad2antrl 728 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u ( X  X.  X ) v )
2614simplbi 460 . . . . . . 7  |-  ( u ( X  X.  X
) v  ->  u  e.  X )
2725, 26syl 17 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u  e.  X
)
2814simprbi 464 . . . . . . 7  |-  ( u ( X  X.  X
) v  ->  v  e.  X )
2925, 28syl 17 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  v  e.  X
)
3027, 29, 16syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u R v  <->  -.  ( u  .<  v  \/  v  .<  u ) ) )
3124, 30mpbid 212 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( u  .<  v  \/  v  .<  u ) )
32 simprr 760 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  v R w )
333brel 4874 . . . . . . . 8  |-  ( v R w  ->  (
v  e.  X  /\  w  e.  X )
)
3433simprd 463 . . . . . . 7  |-  ( v R w  ->  w  e.  X )
3532, 34syl 17 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  w  e.  X
)
361brdifun 7377 . . . . . 6  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v R w  <->  -.  ( v  .<  w  \/  w  .<  v ) ) )
3729, 35, 36syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( v R w  <->  -.  ( v  .<  w  \/  w  .<  v ) ) )
3832, 37mpbid 212 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( v  .<  w  \/  w  .<  v ) )
39 simpl 457 . . . . . . 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 4755 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  X  /\  w  e.  X  /\  v  e.  X ) )  -> 
( u  .<  w  ->  ( u  .<  v  \/  v  .<  w ) ) )
4239, 27, 35, 29, 41syl13anc 1234 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u  .<  w  ->  ( u  .<  v  \/  v  .<  w
) ) )
4340swopolem 4755 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  X  /\  u  e.  X  /\  v  e.  X ) )  -> 
( w  .<  u  ->  ( w  .<  v  \/  v  .<  u ) ) )
4439, 35, 27, 29, 43syl13anc 1234 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( w  .<  u  ->  ( w  .<  v  \/  v  .<  u
) ) )
45 orcom 387 . . . . . . 7  |-  ( ( v  .<  u  \/  w  .<  v )  <->  ( w  .<  v  \/  v  .<  u ) )
4644, 45syl6ibr 229 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( w  .<  u  ->  ( v  .<  u  \/  w  .<  v ) ) )
4742, 46orim12d 841 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( ( u 
.<  w  \/  w  .<  u )  ->  (
( u  .<  v  \/  v  .<  w )  \/  ( v  .<  u  \/  w  .<  v ) ) ) )
48 or4 528 . . . . 5  |-  ( ( ( u  .<  v  \/  v  .<  w )  \/  ( v  .<  u  \/  w  .<  v ) )  <->  ( (
u  .<  v  \/  v  .<  u )  \/  (
v  .<  w  \/  w  .<  v ) ) )
4947, 48syl6ib 228 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( ( u 
.<  w  \/  w  .<  u )  ->  (
( u  .<  v  \/  v  .<  u )  \/  ( v  .<  w  \/  w  .<  v ) ) ) )
5031, 38, 49mtord 660 . . 3  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( u  .<  w  \/  w  .<  u ) )
511brdifun 7377 . . . 4  |-  ( ( u  e.  X  /\  w  e.  X )  ->  ( u R w  <->  -.  ( u  .<  w  \/  w  .<  u ) ) )
5227, 35, 51syl2anc 661 . . 3  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u R w  <->  -.  ( u  .<  w  \/  w  .<  u ) ) )
5350, 52mpbird 234 . 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 4756 . . . . . 6  |-  ( ph  ->  .<  Po  X )
56 poirr 4757 . . . . . 6  |-  ( ( 
.<  Po  X  /\  u  e.  X )  ->  -.  u  .<  u )
5755, 56sylan 471 . . . . 5  |-  ( (
ph  /\  u  e.  X )  ->  -.  u  .<  u )
58 pm1.2 513 . . . . 5  |-  ( ( u  .<  u  \/  u  .<  u )  ->  u  .<  u )
5957, 58nsyl 123 . . . 4  |-  ( (
ph  /\  u  e.  X )  ->  -.  ( u  .<  u  \/  u  .<  u )
)
60 simpr 461 . . . . 5  |-  ( (
ph  /\  u  e.  X )  ->  u  e.  X )
611brdifun 7377 . . . . 5  |-  ( ( u  e.  X  /\  u  e.  X )  ->  ( u R u  <->  -.  ( u  .<  u  \/  u  .<  u ) ) )
6260, 60, 61syl2anc 661 . . . 4  |-  ( (
ph  /\  u  e.  X )  ->  (
u R u  <->  -.  (
u  .<  u  \/  u  .<  u ) ) )
6359, 62mpbird 234 . . 3  |-  ( (
ph  /\  u  e.  X )  ->  u R u )
643ssbri 4439 . . . . 5  |-  ( u R u  ->  u
( X  X.  X
) u )
65 brxp 4856 . . . . . 6  |-  ( u ( X  X.  X
) u  <->  ( u  e.  X  /\  u  e.  X ) )
6665simplbi 460 . . . . 5  |-  ( u ( X  X.  X
) u  ->  u  e.  X )
6764, 66syl 17 . . . 4  |-  ( u R u  ->  u  e.  X )
6867adantl 466 . . 3  |-  ( (
ph  /\  u R u )  ->  u  e.  X )
6963, 68impbida 835 . 2  |-  ( ph  ->  ( u  e.  X  <->  u R u ) )
707, 23, 53, 69iserd 7376 1  |-  ( ph  ->  R  Er  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    \/ wo 368    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    \ cdif 3413    u. cun 3414    C_ wss 3416   class class class wbr 4397    Po wpo 4744    X. cxp 4823   `'ccnv 4824   Rel wrel 4830    Er wer 7347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-po 4746  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-er 7350
This theorem is referenced by: (None)
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