MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swoord1 Unicode version

Theorem swoord1 6705
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 19 . . . 4  |-  ( ph  ->  ph )
2 swoord.6 . . . . 5  |-  ( ph  ->  A R B )
3 swoer.1 . . . . . . 7  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
4 difss 3316 . . . . . . 7  |-  ( ( X  X.  X ) 
\  (  .<  u.  `'  .<  ) )  C_  ( X  X.  X )
53, 4eqsstri 3221 . . . . . 6  |-  R  C_  ( X  X.  X
)
65ssbri 4081 . . . . 5  |-  ( A R B  ->  A
( X  X.  X
) B )
7 df-br 4040 . . . . . 6  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
8 opelxp1 4738 . . . . . 6  |-  ( <. A ,  B >.  e.  ( X  X.  X
)  ->  A  e.  X )
97, 8sylbi 187 . . . . 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 4339 . . . 4  |-  ( (
ph  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X ) )  -> 
( A  .<  C  -> 
( A  .<  B  \/  B  .<  C ) ) )
151, 10, 11, 12, 14syl13anc 1184 . . 3  |-  ( ph  ->  ( A  .<  C  -> 
( A  .<  B  \/  B  .<  C ) ) )
163brdifun 6703 . . . . . . 7  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
1710, 12, 16syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
182, 17mpbid 201 . . . . 5  |-  ( ph  ->  -.  ( A  .<  B  \/  B  .<  A ) )
19 orc 374 . . . . 5  |-  ( A 
.<  B  ->  ( A 
.<  B  \/  B  .<  A ) )
2018, 19nsyl 113 . . . 4  |-  ( ph  ->  -.  A  .<  B )
21 biorf 394 . . . 4  |-  ( -.  A  .<  B  ->  ( B  .<  C  <->  ( A  .<  B  \/  B  .<  C ) ) )
2220, 21syl 15 . . 3  |-  ( ph  ->  ( B  .<  C  <->  ( A  .<  B  \/  B  .<  C ) ) )
2315, 22sylibrd 225 . 2  |-  ( ph  ->  ( A  .<  C  ->  B  .<  C ) )
2413swopolem 4339 . . . 4  |-  ( (
ph  /\  ( B  e.  X  /\  C  e.  X  /\  A  e.  X ) )  -> 
( B  .<  C  -> 
( B  .<  A  \/  A  .<  C ) ) )
251, 12, 11, 10, 24syl13anc 1184 . . 3  |-  ( ph  ->  ( B  .<  C  -> 
( B  .<  A  \/  A  .<  C ) ) )
26 olc 373 . . . . 5  |-  ( B 
.<  A  ->  ( A 
.<  B  \/  B  .<  A ) )
2718, 26nsyl 113 . . . 4  |-  ( ph  ->  -.  B  .<  A )
28 biorf 394 . . . 4  |-  ( -.  B  .<  A  ->  ( A  .<  C  <->  ( B  .<  A  \/  A  .<  C ) ) )
2927, 28syl 15 . . 3  |-  ( ph  ->  ( A  .<  C  <->  ( B  .<  A  \/  A  .<  C ) ) )
3025, 29sylibrd 225 . 2  |-  ( ph  ->  ( B  .<  C  ->  A  .<  C ) )
3123, 30impbid 183 1  |-  ( ph  ->  ( A  .<  C  <->  B  .<  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163   <.cop 3656   class class class wbr 4039    X. cxp 4703   `'ccnv 4704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713
  Copyright terms: Public domain W3C validator