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Theorem sosn 4904
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 3993 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
2 elsni 3993 . . . . . . 7  |-  ( y  e.  { A }  ->  y  =  A )
32eqcomd 2457 . . . . . 6  |-  ( y  e.  { A }  ->  A  =  y )
41, 3sylan9eq 2505 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  x  =  y )
543mix2d 1184 . . . 4  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  (
x R y  \/  x  =  y  \/  y R x ) )
65rgen2a 2815 . . 3  |-  A. x  e.  { A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x )
7 df-so 4756 . . 3  |-  ( R  Or  { A }  <->  ( R  Po  { A }  /\  A. x  e. 
{ A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x ) ) )
86, 7mpbiran2 930 . 2  |-  ( R  Or  { A }  <->  R  Po  { A }
)
9 posn 4903 . 2  |-  ( Rel 
R  ->  ( R  Po  { A }  <->  -.  A R A ) )
108, 9syl5bb 261 1  |-  ( Rel 
R  ->  ( R  Or  { A }  <->  -.  A R A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    \/ w3o 984    e. wcel 1887   A.wral 2737   {csn 3968   class class class wbr 4402    Po wpo 4753    Or wor 4754   Rel wrel 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841
This theorem is referenced by:  wesn  4906
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