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Theorem swoord1 6222
Description: The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)
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 ) ) )
swoord.4  |-  ( ph  ->  B  e.  X )
swoord.5  |-  ( ph  ->  C  e.  X )
swoord.6  |-  ( ph  ->  A R B )
Assertion
Ref Expression
swoord1  |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
Distinct variable groups:    x, y, z, 
.<    x, A, y, z   
x, B, y, z   
x, C, y, z    ph, x, y, z    x, X, y, z
Allowed substitution hints:    R( x, y, z)

Proof of Theorem swoord1
StepHypRef Expression
1 id 19 . . . 4  |-  ( ph  ->  ph )
2 swoord.6 . . . . 5  |-  ( ph  ->  A R B )
3 swoer.1 . . . . . . 7  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
4 difss 3108 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
53, 4eqsstri 3038 . . . . . 6  |-  R  C_  ( X  X.  X
)
65ssbri 3847 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
7 df-br 3806 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
8 opelxp1 4423 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
97, 8sylbi 119 . . . . 5  |-  ( A ( X  X.  X
) B  ->  A  e.  X )
102, 6, 93syl 17 . . . 4  |-  ( ph  ->  A  e.  X )
11 swoord.5 . . . 4  |-  ( ph  ->  C  e.  X )
12 swoord.4 . . . 4  |-  ( ph  ->  B  e.  X )
13 swoer.3 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( x  .<  y  ->  ( x  .<  z  \/  z  .<  y ) ) )
1413swopolem 4088 . . . 4  |-  ( (
ph  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X ) )  -> 
( A  .<  C  -> 
( A  .<  B  \/  B  .<  C ) ) )
151, 10, 11, 12, 14syl13anc 1172 . . 3  |-  ( ph  ->  ( A  .<  C  -> 
( A  .<  B  \/  B  .<  C ) ) )
163brdifun 6220 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1710, 12, 16syl2anc 403 . . . . . 6  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
182, 17mpbid 145 . . . . 5  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
19 orc 666 . . . . 5  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2018, 19nsyl 591 . . . 4  |-  ( ph  ->  -.  A  .<  B )
21 biorf 696 . . . 4  |-  ( -.  A  .<  B  ->  ( B  .<  C  <->  ( A  .<  B  \/  B  .<  C ) ) )
2220, 21syl 14 . . 3  |-  ( ph  ->  ( B  .<  C  <->  ( A  .<  B  \/  B  .<  C ) ) )
2315, 22sylibrd 167 . 2  |-  ( ph  ->  ( A  .<  C  ->  B  .<  C ) )
2413swopolem 4088 . . . 4  |-  ( (
ph  /\  ( B  e.  X  /\  C  e.  X  /\  A  e.  X ) )  -> 
( B  .<  C  -> 
( B  .<  A  \/  A  .<  C ) ) )
251, 12, 11, 10, 24syl13anc 1172 . . 3  |-  ( ph  ->  ( B  .<  C  -> 
( B  .<  A  \/  A  .<  C ) ) )
26 olc 665 . . . . 5  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2718, 26nsyl 591 . . . 4  |-  ( ph  ->  -.  B  .<  A )
28 biorf 696 . . . 4  |-  ( -.  B  .<  A  ->  ( A  .<  C  <->  ( B  .<  A  \/  A  .<  C ) ) )
2927, 28syl 14 . . 3  |-  ( ph  ->  ( A  .<  C  <->  ( B  .<  A  \/  A  .<  C ) ) )
3025, 29sylibrd 167 . 2  |-  ( ph  ->  ( B  .<  C  ->  A  .<  C ) )
3123, 30impbid 127 1  |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434    \ cdif 2979    u. cun 2980   <.cop 3419   class class class wbr 3805    X. cxp 4389   `'ccnv 4390
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399
This theorem is referenced by: (None)
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