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Theorem swoord1 7400
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 23 . . . 4  |-  ( ph  ->  ph )
2 swoord.6 . . . . 5  |-  ( ph  ->  A R B )
3 swoer.1 . . . . . . 7  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
4 difss 3598 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
53, 4eqsstri 3500 . . . . . 6  |-  R  C_  ( X  X.  X
)
65ssbri 4468 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
7 df-br 4427 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
8 opelxp1 4887 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
97, 8sylbi 198 . . . . 5  |-  ( A ( X  X.  X
) B  ->  A  e.  X )
102, 6, 93syl 18 . . . 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 4784 . . . 4  |-  ( (
ph  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X ) )  -> 
( A  .<  C  -> 
( A  .<  B  \/  B  .<  C ) ) )
151, 10, 11, 12, 14syl13anc 1266 . . 3  |-  ( ph  ->  ( A  .<  C  -> 
( A  .<  B  \/  B  .<  C ) ) )
163brdifun 7398 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1710, 12, 16syl2anc 665 . . . . . 6  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
182, 17mpbid 213 . . . . 5  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
19 orc 386 . . . . 5  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2018, 19nsyl 124 . . . 4  |-  ( ph  ->  -.  A  .<  B )
21 biorf 406 . . . 4  |-  ( -.  A  .<  B  ->  ( B  .<  C  <->  ( A  .<  B  \/  B  .<  C ) ) )
2220, 21syl 17 . . 3  |-  ( ph  ->  ( B  .<  C  <->  ( A  .<  B  \/  B  .<  C ) ) )
2315, 22sylibrd 237 . 2  |-  ( ph  ->  ( A  .<  C  ->  B  .<  C ) )
2413swopolem 4784 . . . 4  |-  ( (
ph  /\  ( B  e.  X  /\  C  e.  X  /\  A  e.  X ) )  -> 
( B  .<  C  -> 
( B  .<  A  \/  A  .<  C ) ) )
251, 12, 11, 10, 24syl13anc 1266 . . 3  |-  ( ph  ->  ( B  .<  C  -> 
( B  .<  A  \/  A  .<  C ) ) )
26 olc 385 . . . . 5  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2718, 26nsyl 124 . . . 4  |-  ( ph  ->  -.  B  .<  A )
28 biorf 406 . . . 4  |-  ( -.  B  .<  A  ->  ( A  .<  C  <->  ( B  .<  A  \/  A  .<  C ) ) )
2927, 28syl 17 . . 3  |-  ( ph  ->  ( A  .<  C  <->  ( B  .<  A  \/  A  .<  C ) ) )
3025, 29sylibrd 237 . 2  |-  ( ph  ->  ( B  .<  C  ->  A  .<  C ) )
3123, 30impbid 193 1  |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    \ cdif 3439    u. cun 3440   <.cop 4008   class class class wbr 4426    X. cxp 4852   `'ccnv 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-cnv 4862
This theorem is referenced by: (None)
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