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Theorem sosng 4359
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 4044 . . 3  R  Or  { }  R  Po  { }
2 posng 4358 . . 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 3394 . . . . . . . . . . 11  { }
8 elsni 3394 . . . . . . . . . . 11  { }
97, 8breqan12d 3773 . . . . . . . . . 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 2390 . . . . . 6  R  { }  { }  { }  R  R  R
1413ralrimivva 2398 . . . . 5  R  { }  { } 
{ }  R  R  R
1514adantl 262 . . . 4  Rel  R  _V  R  { }  { }  { }  R  R  R
16 df-iso 4028 . . . 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 629   wcel 1393  wral 2303   _Vcvv 2554   {csn 3370   class class class wbr 3758    Po wpo 4025    Or wor 4026   Rel wrel 4296
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-sbc 2762  df-un 2919  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-po 4027  df-iso 4028
This theorem is referenced by: (None)
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