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Theorem sosn 4775
Description: Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
sosn  |-  ( Rel 
R  ->  ( R  Or  { A }  <->  -.  A R A ) )

Proof of Theorem sosn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsni 3677 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
2 elsni 3677 . . . . . . 7  |-  ( y  e.  { A }  ->  y  =  A )
32eqcomd 2301 . . . . . 6  |-  ( y  e.  { A }  ->  A  =  y )
41, 3sylan9eq 2348 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  x  =  y )
5 3mix2 1125 . . . . 5  |-  ( x  =  y  ->  (
x R y  \/  x  =  y  \/  y R x ) )
64, 5syl 15 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x R y  \/  x  =  y  \/  y R x ) )
76rgen2a 2622 . . 3  |-  A. x  e.  { A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x )
8 df-so 4331 . . 3  |-  ( R  Or  { A }  <->  ( R  Po  { A }  /\  A. x  e. 
{ A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x ) ) )
97, 8mpbiran2 885 . 2  |-  ( R  Or  { A }  <->  R  Po  { A }
)
10 posn 4774 . 2  |-  ( Rel 
R  ->  ( R  Po  { A }  <->  -.  A R A ) )
119, 10syl5bb 248 1  |-  ( Rel 
R  ->  ( R  Or  { A }  <->  -.  A R A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653   class class class wbr 4039    Po wpo 4328    Or wor 4329   Rel wrel 4710
This theorem is referenced by:  wesn  4777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712
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