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Theorem swoord1 6071
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  X  X  .<  .<
swoer.3  X  X  X  .<  .<  .<
swoord.4  X
swoord.5  C  X
swoord.6  R
Assertion
Ref Expression
swoord1  .<  C  .<  C
Distinct variable groups:   ,,, 
.<   ,,,   ,,,   , C,,   ,,,   , X,,
Allowed substitution hints:    R(,,)

Proof of Theorem swoord1
StepHypRef Expression
1 id 19 . . . 4
2 swoord.6 . . . . 5  R
3 swoer.1 . . . . . . 7  R  X  X.  X  \  .<  u.  `'  .<
4 difss 3064 . . . . . . 7  X  X.  X 
\  .<  u.  `'  .<  C_  X  X.  X
53, 4eqsstri 2969 . . . . . 6  R  C_  X  X.  X
65ssbri 3797 . . . . 5  R  X  X.  X
7 df-br 3756 . . . . . 6  X  X.  X  <. ,  >.  X  X.  X
8 opelxp1 4320 . . . . . 6  <. ,  >.  X  X.  X  X
97, 8sylbi 114 . . . . 5  X  X.  X  X
102, 6, 93syl 17 . . . 4  X
11 swoord.5 . . . 4  C  X
12 swoord.4 . . . 4  X
13 swoer.3 . . . . 5  X  X  X  .<  .<  .<
1413swopolem 4033 . . . 4  X  C  X  X 
.<  C  .<  .<  C
151, 10, 11, 12, 14syl13anc 1136 . . 3  .<  C  .<  .<  C
163brdifun 6069 . . . . . . 7  X  X  R  .<  .<
1710, 12, 16syl2anc 391 . . . . . 6  R  .<  .<
182, 17mpbid 135 . . . . 5  .<  .<
19 orc 632 . . . . 5 
.< 
.<  .<
2018, 19nsyl 558 . . . 4  .<
21 biorf 662 . . . 4  .<  .<  C 
.<  .<  C
2220, 21syl 14 . . 3  .<  C 
.<  .<  C
2315, 22sylibrd 158 . 2  .<  C  .<  C
2413swopolem 4033 . . . 4  X  C  X  X 
.<  C  .<  .<  C
251, 12, 11, 10, 24syl13anc 1136 . . 3  .<  C  .<  .<  C
26 olc 631 . . . . 5 
.< 
.<  .<
2718, 26nsyl 558 . . . 4  .<
28 biorf 662 . . . 4  .<  .<  C 
.<  .<  C
2927, 28syl 14 . . 3  .<  C 
.<  .<  C
3025, 29sylibrd 158 . 2  .<  C  .<  C
3123, 30impbid 120 1  .<  C  .<  C
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   w3a 884   wceq 1242   wcel 1390    \ cdif 2908    u. cun 2909   <.cop 3370   class class class wbr 3755    X. cxp 4286   `'ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296
This theorem is referenced by: (None)
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