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Theorem swoer 6562
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 3261 . . . . 5  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
31, 2eqsstri 3187 . . . 4  |-  R  C_  ( X  X.  X
)
4 relxp 4735 . . . 4  |-  Rel  ( X  X.  X )
5 relss 4713 . . . 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 110 . . 3  |-  ( (
ph  /\  u R
v )  ->  u R v )
9 orcom 728 . . . . . 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 667 . . . 4  |-  ( (
ph  /\  u R
v )  ->  ( -.  ( u  .<  v  \/  v  .<  u )  <->  -.  ( v  .<  u  \/  u  .<  v ) ) )
123ssbri 4047 . . . . . . 7  |-  ( u R v  ->  u
( X  X.  X
) v )
1312adantl 277 . . . . . 6  |-  ( (
ph  /\  u R
v )  ->  u
( X  X.  X
) v )
14 brxp 4657 . . . . . 6  |-  ( u ( X  X.  X
) v  <->  ( u  e.  X  /\  v  e.  X ) )
1513, 14sylib 122 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  (
u  e.  X  /\  v  e.  X )
)
161brdifun 6561 . . . . 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 114 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  v  e.  X )
1915simpld 112 . . . . 5  |-  ( (
ph  /\  u R
v )  ->  u  e.  X )
201brdifun 6561 . . . . 5  |-  ( ( v  e.  X  /\  u  e.  X )  ->  ( v R u  <->  -.  ( v  .<  u  \/  u  .<  v ) ) )
2118, 19, 20syl2anc 411 . . . 4  |-  ( (
ph  /\  u R
v )  ->  (
v R u  <->  -.  (
v  .<  u  \/  u  .<  v ) ) )
2211, 17, 213bitr4d 220 . . 3  |-  ( (
ph  /\  u R
v )  ->  (
u R v  <->  v R u ) )
238, 22mpbid 147 . 2  |-  ( (
ph  /\  u R
v )  ->  v R u )
24 simprl 529 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u R v )
2512ad2antrl 490 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u ( X  X.  X ) v )
2614simplbi 274 . . . . . . 7  |-  ( u ( X  X.  X
) v  ->  u  e.  X )
2725, 26syl 14 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  u  e.  X
)
2814simprbi 275 . . . . . . 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 411 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u R v  <->  -.  ( u  .<  v  \/  v  .<  u ) ) )
3124, 30mpbid 147 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( u  .<  v  \/  v  .<  u ) )
32 simprr 531 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  v R w )
333brel 4678 . . . . . . . 8  |-  ( v R w  ->  (
v  e.  X  /\  w  e.  X )
)
3433simprd 114 . . . . . . 7  |-  ( v R w  ->  w  e.  X )
3532, 34syl 14 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  w  e.  X
)
361brdifun 6561 . . . . . 6  |-  ( ( v  e.  X  /\  w  e.  X )  ->  ( v R w  <->  -.  ( v  .<  w  \/  w  .<  v ) ) )
3729, 35, 36syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( v R w  <->  -.  ( v  .<  w  \/  w  .<  v ) ) )
3832, 37mpbid 147 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( v  .<  w  \/  w  .<  v ) )
39 simpl 109 . . . . . . 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 4305 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  X  /\  w  e.  X  /\  v  e.  X ) )  -> 
( u  .<  w  ->  ( u  .<  v  \/  v  .<  w ) ) )
4239, 27, 35, 29, 41syl13anc 1240 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u  .<  w  ->  ( u  .<  v  \/  v  .<  w
) ) )
4340swopolem 4305 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  X  /\  u  e.  X  /\  v  e.  X ) )  -> 
( w  .<  u  ->  ( w  .<  v  \/  v  .<  u ) ) )
4439, 35, 27, 29, 43syl13anc 1240 . . . . . . 7  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( w  .<  u  ->  ( w  .<  v  \/  v  .<  u
) ) )
45 orcom 728 . . . . . . 7  |-  ( ( v  .<  u  \/  w  .<  v )  <->  ( w  .<  v  \/  v  .<  u ) )
4644, 45imbitrrdi 162 . . . . . 6  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( w  .<  u  ->  ( v  .<  u  \/  w  .<  v ) ) )
4742, 46orim12d 786 . . . . 5  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( ( u 
.<  w  \/  w  .<  u )  ->  (
( u  .<  v  \/  v  .<  w )  \/  ( v  .<  u  \/  w  .<  v ) ) ) )
48 or4 771 . . . . 5  |-  ( ( ( u  .<  v  \/  v  .<  w )  \/  ( v  .<  u  \/  w  .<  v ) )  <->  ( (
u  .<  v  \/  v  .<  u )  \/  (
v  .<  w  \/  w  .<  v ) ) )
4947, 48imbitrdi 161 . . . 4  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( ( u 
.<  w  \/  w  .<  u )  ->  (
( u  .<  v  \/  v  .<  u )  \/  ( v  .<  w  \/  w  .<  v ) ) ) )
5031, 38, 49mtord 783 . . 3  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  -.  ( u  .<  w  \/  w  .<  u ) )
511brdifun 6561 . . . 4  |-  ( ( u  e.  X  /\  w  e.  X )  ->  ( u R w  <->  -.  ( u  .<  w  \/  w  .<  u ) ) )
5227, 35, 51syl2anc 411 . . 3  |-  ( (
ph  /\  ( u R v  /\  v R w ) )  ->  ( u R w  <->  -.  ( u  .<  w  \/  w  .<  u ) ) )
5350, 52mpbird 167 . 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 4306 . . . . . 6  |-  ( ph  ->  .<  Po  X )
56 poirr 4307 . . . . . 6  |-  ( ( 
.<  Po  X  /\  u  e.  X )  ->  -.  u  .<  u )
5755, 56sylan 283 . . . . 5  |-  ( (
ph  /\  u  e.  X )  ->  -.  u  .<  u )
58 pm1.2 756 . . . . 5  |-  ( ( u  .<  u  \/  u  .<  u )  ->  u  .<  u )
5957, 58nsyl 628 . . . 4  |-  ( (
ph  /\  u  e.  X )  ->  -.  ( u  .<  u  \/  u  .<  u )
)
60 simpr 110 . . . . 5  |-  ( (
ph  /\  u  e.  X )  ->  u  e.  X )
611brdifun 6561 . . . . 5  |-  ( ( u  e.  X  /\  u  e.  X )  ->  ( u R u  <->  -.  ( u  .<  u  \/  u  .<  u ) ) )
6260, 60, 61syl2anc 411 . . . 4  |-  ( (
ph  /\  u  e.  X )  ->  (
u R u  <->  -.  (
u  .<  u  \/  u  .<  u ) ) )
6359, 62mpbird 167 . . 3  |-  ( (
ph  /\  u  e.  X )  ->  u R u )
643ssbri 4047 . . . . 5  |-  ( u R u  ->  u
( X  X.  X
) u )
65 brxp 4657 . . . . . 6  |-  ( u ( X  X.  X
) u  <->  ( u  e.  X  /\  u  e.  X ) )
6665simplbi 274 . . . . 5  |-  ( u ( X  X.  X
) u  ->  u  e.  X )
6764, 66syl 14 . . . 4  |-  ( u R u  ->  u  e.  X )
6867adantl 277 . . 3  |-  ( (
ph  /\  u R u )  ->  u  e.  X )
6963, 68impbida 596 . 2  |-  ( ph  ->  ( u  e.  X  <->  u R u ) )
707, 23, 53, 69iserd 6560 1  |-  ( ph  ->  R  Er  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148    \ cdif 3126    u. cun 3127    C_ wss 3129   class class class wbr 4003    Po wpo 4294    X. cxp 4624   `'ccnv 4625   Rel wrel 4631    Er wer 6531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-po 4296  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-er 6534
This theorem is referenced by: (None)
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