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Theorem swoord2 6195
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
swoord2  |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
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 swoord2
StepHypRef Expression
1 id 19 . . . 4  |-  ( ph  ->  ph )
2 swoord.5 . . . 4  |-  ( ph  ->  C  e.  X )
3 swoord.6 . . . . 5  |-  ( ph  ->  A R B )
4 swoer.1 . . . . . . 7  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
5 difss 3099 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
64, 5eqsstri 3030 . . . . . 6  |-  R  C_  ( X  X.  X
)
76ssbri 3829 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
8 df-br 3788 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
9 opelxp1 4397 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
108, 9sylbi 119 . . . . 5  |-  ( A ( X  X.  X
) B  ->  A  e.  X )
113, 7, 103syl 17 . . . 4  |-  ( ph  ->  A  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 4062 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  -> 
( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
151, 2, 11, 12, 14syl13anc 1172 . . 3  |-  ( ph  ->  ( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
16 idd 21 . . . 4  |-  ( ph  ->  ( C  .<  B  ->  C  .<  B ) )
174brdifun 6192 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1811, 12, 17syl2anc 403 . . . . . . 7  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
193, 18mpbid 145 . . . . . 6  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
20 olc 665 . . . . . 6  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2119, 20nsyl 591 . . . . 5  |-  ( ph  ->  -.  B  .<  A )
2221pm2.21d 582 . . . 4  |-  ( ph  ->  ( B  .<  A  ->  C  .<  B ) )
2316, 22jaod 670 . . 3  |-  ( ph  ->  ( ( C  .<  B  \/  B  .<  A )  ->  C  .<  B ) )
2415, 23syld 44 . 2  |-  ( ph  ->  ( C  .<  A  ->  C  .<  B ) )
2513swopolem 4062 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X ) )  -> 
( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
261, 2, 12, 11, 25syl13anc 1172 . . 3  |-  ( ph  ->  ( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
27 idd 21 . . . 4  |-  ( ph  ->  ( C  .<  A  ->  C  .<  A ) )
28 orc 666 . . . . . 6  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2919, 28nsyl 591 . . . . 5  |-  ( ph  ->  -.  A  .<  B )
3029pm2.21d 582 . . . 4  |-  ( ph  ->  ( A  .<  B  ->  C  .<  A ) )
3127, 30jaod 670 . . 3  |-  ( ph  ->  ( ( C  .<  A  \/  A  .<  B )  ->  C  .<  A ) )
3226, 31syld 44 . 2  |-  ( ph  ->  ( C  .<  B  ->  C  .<  A ) )
3324, 32impbid 127 1  |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
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 2971    u. cun 2972   <.cop 3403   class class class wbr 3787    X. cxp 4363   `'ccnv 4364
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 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373
This theorem is referenced by: (None)
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