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Theorem sosn 5075
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 4058 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
2 elsni 4058 . . . . . . 7  |-  ( y  e.  { A }  ->  y  =  A )
32eqcomd 2475 . . . . . 6  |-  ( y  e.  { A }  ->  A  =  y )
41, 3sylan9eq 2528 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  x  =  y )
5 3mix2 1166 . . . . 5  |-  ( x  =  y  ->  (
x R y  \/  x  =  y  \/  y R x ) )
64, 5syl 16 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x R y  \/  x  =  y  \/  y R x ) )
76rgen2a 2894 . . 3  |-  A. x  e.  { A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x )
8 df-so 4807 . . 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 917 . 2  |-  ( R  Or  { A }  <->  R  Po  { A }
)
10 posn 5074 . 2  |-  ( Rel 
R  ->  ( R  Po  { A }  <->  -.  A R A ) )
119, 10syl5bb 257 1  |-  ( Rel 
R  ->  ( R  Or  { A }  <->  -.  A R A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 972    e. wcel 1767   A.wral 2817   {csn 4033   class class class wbr 4453    Po wpo 4804    Or wor 4805   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012
This theorem is referenced by:  wesn  5077
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