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Theorem sosn 4913
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 3907 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
2 elsni 3907 . . . . . . 7  |-  ( y  e.  { A }  ->  y  =  A )
32eqcomd 2448 . . . . . 6  |-  ( y  e.  { A }  ->  A  =  y )
41, 3sylan9eq 2495 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A } )  ->  x  =  y )
5 3mix2 1158 . . . . 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 2787 . . 3  |-  A. x  e.  { A } A. y  e.  { A }  ( x R y  \/  x  =  y  \/  y R x )
8 df-so 4647 . . 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 910 . 2  |-  ( R  Or  { A }  <->  R  Po  { A }
)
10 posn 4912 . 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 964    e. wcel 1756   A.wral 2720   {csn 3882   class class class wbr 4297    Po wpo 4644    Or wor 4645   Rel wrel 4850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852
This theorem is referenced by:  wesn  4915
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