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Theorem swoord2 7333
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 22 . . . 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 3626 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
64, 5eqsstri 3529 . . . . . 6  |-  R  C_  ( X  X.  X
)
76ssbri 4484 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
8 df-br 4443 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
9 opelxp1 5026 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
108, 9sylbi 195 . . . . 5  |-  ( A ( X  X.  X
) B  ->  A  e.  X )
113, 7, 103syl 20 . . . 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 4804 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X ) )  -> 
( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
151, 2, 11, 12, 14syl13anc 1225 . . 3  |-  ( ph  ->  ( C  .<  A  -> 
( C  .<  B  \/  B  .<  A ) ) )
16 idd 24 . . . 4  |-  ( ph  ->  ( C  .<  B  ->  C  .<  B ) )
174brdifun 7330 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1811, 12, 17syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
193, 18mpbid 210 . . . . . 6  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
20 olc 384 . . . . . 6  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2119, 20nsyl 121 . . . . 5  |-  ( ph  ->  -.  B  .<  A )
2221pm2.21d 106 . . . 4  |-  ( ph  ->  ( B  .<  A  ->  C  .<  B ) )
2316, 22jaod 380 . . 3  |-  ( ph  ->  ( ( C  .<  B  \/  B  .<  A )  ->  C  .<  B ) )
2415, 23syld 44 . 2  |-  ( ph  ->  ( C  .<  A  ->  C  .<  B ) )
2513swopolem 4804 . . . 4  |-  ( (
ph  /\  ( C  e.  X  /\  B  e.  X  /\  A  e.  X ) )  -> 
( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
261, 2, 12, 11, 25syl13anc 1225 . . 3  |-  ( ph  ->  ( C  .<  B  -> 
( C  .<  A  \/  A  .<  B ) ) )
27 idd 24 . . . 4  |-  ( ph  ->  ( C  .<  A  ->  C  .<  A ) )
28 orc 385 . . . . . 6  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2919, 28nsyl 121 . . . . 5  |-  ( ph  ->  -.  A  .<  B )
3029pm2.21d 106 . . . 4  |-  ( ph  ->  ( A  .<  B  ->  C  .<  A ) )
3127, 30jaod 380 . . 3  |-  ( ph  ->  ( ( C  .<  A  \/  A  .<  B )  ->  C  .<  A ) )
3226, 31syld 44 . 2  |-  ( ph  ->  ( C  .<  B  ->  C  .<  A ) )
3324, 32impbid 191 1  |-  ( ph  ->  ( C  .<  A  <->  C  .<  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    \ cdif 3468    u. cun 3469   <.cop 4028   class class class wbr 4442    X. cxp 4992   `'ccnv 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002
This theorem is referenced by: (None)
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