MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wesn Structured version   Unicode version

Theorem wesn 5057
Description: Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
wesn  |-  ( Rel 
R  ->  ( R  We  { A }  <->  -.  A R A ) )

Proof of Theorem wesn
StepHypRef Expression
1 frsn 5056 . . 3  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )
2 sosn 5055 . . 3  |-  ( Rel 
R  ->  ( R  Or  { A }  <->  -.  A R A ) )
31, 2anbi12d 710 . 2  |-  ( Rel 
R  ->  ( ( R  Fr  { A }  /\  R  Or  { A } )  <->  ( -.  A R A  /\  -.  A R A ) ) )
4 df-we 4826 . 2  |-  ( R  We  { A }  <->  ( R  Fr  { A }  /\  R  Or  { A } ) )
5 pm4.24 643 . 2  |-  ( -.  A R A  <->  ( -.  A R A  /\  -.  A R A ) )
63, 4, 53bitr4g 288 1  |-  ( Rel 
R  ->  ( R  We  { A }  <->  -.  A R A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   {csn 4010   class class class wbr 4433    Or wor 4785    Fr wfr 4821    We wwe 4823   Rel wrel 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-xp 4991  df-rel 4992
This theorem is referenced by:  0we1  7154  canthwe  9027
  Copyright terms: Public domain W3C validator