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Theorem swoord2 6690
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 3303 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
64, 5eqsstri 3208 . . . . . 6  |-  R  C_  ( X  X.  X
)
76ssbri 4065 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
8 df-br 4024 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
9 opelxp1 4722 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
108, 9sylbi 187 . . . . 5  |-  ( A ( X  X.  X
) B  ->  A  e.  X )
113, 7, 103syl 18 . . . 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 4323 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  -> 
( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
151, 2, 11, 12, 14syl13anc 1184 . . 3  |-  ( ph  ->  ( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
16 idd 21 . . . 4  |-  ( ph  ->  ( C  .<  B  ->  C  .<  B ) )
174brdifun 6687 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1811, 12, 17syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
193, 18mpbid 201 . . . . . 6  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
20 olc 373 . . . . . 6  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2119, 20nsyl 113 . . . . 5  |-  ( ph  ->  -.  B  .<  A )
2221pm2.21d 98 . . . 4  |-  ( ph  ->  ( B  .<  A  ->  C  .<  B ) )
2316, 22jaod 369 . . 3  |-  ( ph  ->  ( ( C  .<  B  \/  B  .<  A )  ->  C  .<  B ) )
2415, 23syld 40 . 2  |-  ( ph  ->  ( C  .<  A  ->  C  .<  B ) )
2513swopolem 4323 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X ) )  -> 
( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
261, 2, 12, 11, 25syl13anc 1184 . . 3  |-  ( ph  ->  ( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
27 idd 21 . . . 4  |-  ( ph  ->  ( C  .<  A  ->  C  .<  A ) )
28 orc 374 . . . . . 6  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2919, 28nsyl 113 . . . . 5  |-  ( ph  ->  -.  A  .<  B )
3029pm2.21d 98 . . . 4  |-  ( ph  ->  ( A  .<  B  ->  C  .<  A ) )
3127, 30jaod 369 . . 3  |-  ( ph  ->  ( ( C  .<  A  \/  A  .<  B )  ->  C  .<  A ) )
3226, 31syld 40 . 2  |-  ( ph  ->  ( C  .<  B  ->  C  .<  A ) )
3324, 32impbid 183 1  |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150   <.cop 3643   class class class wbr 4023    X. cxp 4687   `'ccnv 4688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697
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