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Theorem sosng 4356
Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
Assertion
Ref Expression
sosng  Rel  R  _V  R  Or  { }  R

Proof of Theorem sosng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4041 . . 3  R  Or  { }  R  Po  { }
2 posng 4355 . . 3  Rel  R  _V  R  Po  { }  R
31, 2syl5ib 143 . 2  Rel  R  _V  R  Or  { }  R
42biimpar 281 . . . 4  Rel  R  _V  R  R  Po  { }
5 ax-in2 545 . . . . . . . . 9  R  R  R  R
65adantr 261 . . . . . . . 8  R  { }  { }  R  R  R
7 elsni 3391 . . . . . . . . . . 11  { }
8 elsni 3391 . . . . . . . . . . 11  { }
97, 8breqan12d 3770 . . . . . . . . . 10  { }  { }  R  R
109imbi1d 220 . . . . . . . . 9  { }  { }  R  R  R  R  R  R
1110adantl 262 . . . . . . . 8  R  { }  { }  R  R  R  R  R  R
126, 11mpbird 156 . . . . . . 7  R  { }  { }  R  R  R
1312ralrimivw 2387 . . . . . 6  R  { }  { }  { }  R  R  R
1413ralrimivva 2395 . . . . 5  R  { }  { } 
{ }  R  R  R
1514adantl 262 . . . 4  Rel  R  _V  R  { }  { }  { }  R  R  R
16 df-iso 4025 . . . 4  R  Or  { }  R  Po  { } 
{ }  { }  { }  R  R  R
174, 15, 16sylanbrc 394 . . 3  Rel  R  _V  R  R  Or  { }
1817ex 108 . 2  Rel  R  _V  R  R  Or  { }
193, 18impbid 120 1  Rel  R  _V  R  Or  { }  R
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   wcel 1390  wral 2300   _Vcvv 2551   {csn 3367   class class class wbr 3755    Po wpo 4022    Or wor 4023   Rel wrel 4293
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-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-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024  df-iso 4025
This theorem is referenced by: (None)
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