ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sosng Unicode version

Theorem sosng 4582
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 4205 . . 3  |-  ( R  Or  { A }  ->  R  Po  { A } )
2 posng 4581 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Po  { A } 
<->  -.  A R A ) )
31, 2syl5ib 153 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Or  { A }  ->  -.  A R A ) )
42biimpar 295 . . . 4  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  -.  A R A )  ->  R  Po  { A } )
5 ax-in2 589 . . . . . . . . 9  |-  ( -.  A R A  -> 
( A R A  ->  ( x R z  \/  z R y ) ) )
65adantr 274 . . . . . . . 8  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  -> 
( A R A  ->  ( x R z  \/  z R y ) ) )
7 elsni 3515 . . . . . . . . . . 11  |-  ( x  e.  { A }  ->  x  =  A )
8 elsni 3515 . . . . . . . . . . 11  |-  ( y  e.  { A }  ->  y  =  A )
97, 8breqan12d 3915 . . . . . . . . . 10  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x R y  <->  A R A ) )
109imbi1d 230 . . . . . . . . 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 275 . . . . . . . 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 166 . . . . . . 7  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  -> 
( x R y  ->  ( x R z  \/  z R y ) ) )
1312ralrimivw 2483 . . . . . 6  |-  ( ( -.  A R A  /\  ( x  e. 
{ A }  /\  y  e.  { A } ) )  ->  A. z  e.  { A }  ( x R y  ->  ( x R z  \/  z R y ) ) )
1413ralrimivva 2491 . . . . 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 275 . . . 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 4189 . . . 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 413 . . 3  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  -.  A R A )  ->  R  Or  { A } )
1817ex 114 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( -.  A R A  ->  R  Or  { A } ) )
193, 18impbid 128 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 103    <-> wb 104    \/ wo 682    e. wcel 1465   A.wral 2393   _Vcvv 2660   {csn 3497   class class class wbr 3899    Po wpo 4186    Or wor 4187   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-sbc 2883  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188  df-iso 4189
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator