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Theorem sosng 4459
Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
Assertion
Ref Expression
sosng  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Or  { A } 
<->  -.  A R A ) )

Proof of Theorem sosng
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4096 . . 3  |-  ( R  Or  { A }  ->  R  Po  { A } )
2 posng 4458 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Po  { A } 
<->  -.  A R A ) )
31, 2syl5ib 152 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Or  { A }  ->  -.  A R A ) )
42biimpar 291 . . . 4  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  -.  A R A )  ->  R  Po  { A } )
5 ax-in2 578 . . . . . . . . 9  |-  ( -.  A R A  -> 
( A R A  ->  ( x R z  \/  z R y ) ) )
65adantr 270 . . . . . . . 8  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  -> 
( A R A  ->  ( x R z  \/  z R y ) ) )
7 elsni 3434 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  x  =  A )
8 elsni 3434 . . . . . . . . . . 11  |-  ( y  e.  { A }  ->  y  =  A )
97, 8breqan12d 3820 . . . . . . . . . 10  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x R y  <->  A R A ) )
109imbi1d 229 . . . . . . . . 9  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
( x R y  ->  ( x R z  \/  z R y ) )  <->  ( A R A  ->  ( x R z  \/  z R y ) ) ) )
1110adantl 271 . . . . . . . 8  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  -> 
( ( x R y  ->  ( x R z  \/  z R y ) )  <-> 
( A R A  ->  ( x R z  \/  z R y ) ) ) )
126, 11mpbird 165 . . . . . . 7  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  -> 
( x R y  ->  ( x R z  \/  z R y ) ) )
1312ralrimivw 2440 . . . . . 6  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  ->  A. z  e.  { A }  ( x R y  ->  ( x R z  \/  z R y ) ) )
1413ralrimivva 2448 . . . . 5  |-  ( -.  A R A  ->  A. x  e.  { A } A. y  e.  { A } A. z  e. 
{ A }  (
x R y  -> 
( x R z  \/  z R y ) ) )
1514adantl 271 . . . 4  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  -.  A R A )  ->  A. x  e.  { A } A. y  e.  { A } A. z  e.  { A }  ( x R y  ->  (
x R z  \/  z R y ) ) )
16 df-iso 4080 . . . 4  |-  ( R  Or  { A }  <->  ( R  Po  { A }  /\  A. x  e. 
{ A } A. y  e.  { A } A. z  e.  { A }  ( x R y  ->  (
x R z  \/  z R y ) ) ) )
174, 15, 16sylanbrc 408 . . 3  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  -.  A R A )  ->  R  Or  { A } )
1817ex 113 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( -.  A R A  ->  R  Or  { A } ) )
193, 18impbid 127 1  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Or  { A } 
<->  -.  A R A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    e. wcel 1434   A.wral 2353   _Vcvv 2610   {csn 3416   class class class wbr 3805    Po wpo 4077    Or wor 4078   Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-po 4079  df-iso 4080
This theorem is referenced by: (None)
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