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Theorem soss 4042
Description: Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
soss 
C_  R  Or  R  Or

Proof of Theorem soss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poss 4026 . . 3 
C_  R  Po  R  Po
2 ssel 2933 . . . . . . . 8 
C_
3 ssel 2933 . . . . . . . 8 
C_
4 ssel 2933 . . . . . . . 8 
C_
52, 3, 43anim123d 1213 . . . . . . 7 
C_
65imim1d 69 . . . . . 6 
C_  R  R  R  R  R  R
762alimdv 1758 . . . . 5 
C_  R  R  R  R  R  R
87alimdv 1756 . . . 4 
C_  R  R  R  R  R  R
9 r3al 2360 . . . 4  R  R  R  R  R  R
10 r3al 2360 . . . 4  R  R  R  R  R  R
118, 9, 103imtr4g 194 . . 3 
C_  R  R  R  R  R  R
121, 11anim12d 318 . 2 
C_  R  Po  R  R  R  R  Po  R  R  R
13 df-iso 4025 . 2  R  Or  R  Po  R  R  R
14 df-iso 4025 . 2  R  Or  R  Po  R  R  R
1512, 13, 143imtr4g 194 1 
C_  R  Or  R  Or
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wo 628   w3a 884  wal 1240   wcel 1390  wral 2300    C_ wss 2911   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-in 2918  df-ss 2925  df-po 4024  df-iso 4025
This theorem is referenced by:  soeq2  4044
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