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Theorem swoord2 6938
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 21 . . . 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 3476 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
64, 5eqsstri 3380 . . . . . 6  |-  R  C_  ( X  X.  X
)
76ssbri 4257 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
8 df-br 4216 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
9 opelxp1 4914 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
108, 9sylbi 189 . . . . 5  |-  ( A ( X  X.  X
) B  ->  A  e.  X )
113, 7, 103syl 19 . . . 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 4515 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  -> 
( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
151, 2, 11, 12, 14syl13anc 1187 . . 3  |-  ( ph  ->  ( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
16 idd 23 . . . 4  |-  ( ph  ->  ( C  .<  B  ->  C  .<  B ) )
174brdifun 6935 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1811, 12, 17syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
193, 18mpbid 203 . . . . . 6  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
20 olc 375 . . . . . 6  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2119, 20nsyl 116 . . . . 5  |-  ( ph  ->  -.  B  .<  A )
2221pm2.21d 101 . . . 4  |-  ( ph  ->  ( B  .<  A  ->  C  .<  B ) )
2316, 22jaod 371 . . 3  |-  ( ph  ->  ( ( C  .<  B  \/  B  .<  A )  ->  C  .<  B ) )
2415, 23syld 43 . 2  |-  ( ph  ->  ( C  .<  A  ->  C  .<  B ) )
2513swopolem 4515 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X ) )  -> 
( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
261, 2, 12, 11, 25syl13anc 1187 . . 3  |-  ( ph  ->  ( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
27 idd 23 . . . 4  |-  ( ph  ->  ( C  .<  A  ->  C  .<  A ) )
28 orc 376 . . . . . 6  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2919, 28nsyl 116 . . . . 5  |-  ( ph  ->  -.  A  .<  B )
3029pm2.21d 101 . . . 4  |-  ( ph  ->  ( A  .<  B  ->  C  .<  A ) )
3127, 30jaod 371 . . 3  |-  ( ph  ->  ( ( C  .<  A  \/  A  .<  B )  ->  C  .<  A ) )
3226, 31syld 43 . 2  |-  ( ph  ->  ( C  .<  B  ->  C  .<  A ) )
3324, 32impbid 185 1  |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319    u. cun 3320   <.cop 3819   class class class wbr 4215    X. cxp 4879   `'ccnv 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889
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