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Theorem sowlin 4048
Description: A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
Assertion
Ref Expression
sowlin  R  Or  C  D  R C  R D  D R C

Proof of Theorem sowlin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3758 . . . . 5  R  R
2 breq1 3758 . . . . . 6  R  R
32orbi1d 704 . . . . 5  R  R  R  R
41, 3imbi12d 223 . . . 4  R  R  R  R  R  R
54imbi2d 219 . . 3  R  Or  R  R  R  R  Or  R  R  R
6 breq2 3759 . . . . 5  C  R  R C
7 breq2 3759 . . . . . 6  C  R  R C
87orbi2d 703 . . . . 5  C  R  R  R  R C
96, 8imbi12d 223 . . . 4  C  R  R  R  R C  R  R C
109imbi2d 219 . . 3  C  R  Or  R  R  R  R  Or  R C  R  R C
11 breq2 3759 . . . . . 6  D  R  R D
12 breq1 3758 . . . . . 6  D  R C  D R C
1311, 12orbi12d 706 . . . . 5  D  R  R C  R D  D R C
1413imbi2d 219 . . . 4  D  R C  R  R C  R C  R D  D R C
1514imbi2d 219 . . 3  D  R  Or  R C  R  R C  R  Or  R C  R D  D R C
16 df-iso 4025 . . . . 5  R  Or  R  Po  R  R  R
17 3anass 888 . . . . . . 7
18 rsp 2363 . . . . . . . . 9  R  R  R  R  R  R
19 rsp2 2365 . . . . . . . . 9  R  R  R  R  R  R
2018, 19syl6 29 . . . . . . . 8  R  R  R  R  R  R
2120impd 242 . . . . . . 7  R  R  R  R  R  R
2217, 21syl5bi 141 . . . . . 6  R  R  R  R  R  R
2322adantl 262 . . . . 5  R  Po  R  R  R  R  R  R
2416, 23sylbi 114 . . . 4  R  Or  R  R  R
2524com12 27 . . 3  R  Or  R  R  R
265, 10, 15, 25vtocl3ga 2617 . 2  C  D  R  Or  R C  R D  D R C
2726impcom 116 1  R  Or  C  D  R C  R D  D R C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wo 628   w3a 884   wceq 1242   wcel 1390  wral 2300   class class class wbr 3755    Po wpo 4022    Or wor 4023
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-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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-iso 4025
This theorem is referenced by:  sotri2  4665  sotri3  4666  addextpr  6593  cauappcvgprlemloc  6624  caucvgprlemloc  6646  ltsosr  6692  axpre-ltwlin  6767  xrlelttr  8492  xrltletr  8493  xrletr  8494
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