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Theorem sosn 4759
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 3664 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
2 elsni 3664 . . . . . . 7  |-  ( y  e.  { A }  ->  y  =  A )
32eqcomd 2288 . . . . . 6  |-  ( y  e.  { A }  ->  A  =  y )
41, 3sylan9eq 2335 . . . . 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 2609 . . 3  |-  A. x  e.  { A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x )
8 df-so 4315 . . 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 4758 . 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 1623    e. wcel 1684   A.wral 2543   {csn 3640   class class class wbr 4023    Po wpo 4312    Or wor 4313   Rel wrel 4694
This theorem is referenced by:  wesn  4761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696
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