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

Theorem tprot 3899
Description: Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tprot  |-  { A ,  B ,  C }  =  { B ,  C ,  A }

Proof of Theorem tprot
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 3orrot 942 . . 3  |-  ( ( x  =  A  \/  x  =  B  \/  x  =  C )  <->  ( x  =  B  \/  x  =  C  \/  x  =  A )
)
21abbii 2548 . 2  |-  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }  =  { x  |  (
x  =  B  \/  x  =  C  \/  x  =  A ) }
3 dftp2 3854 . 2  |-  { A ,  B ,  C }  =  { x  |  ( x  =  A  \/  x  =  B  \/  x  =  C ) }
4 dftp2 3854 . 2  |-  { B ,  C ,  A }  =  { x  |  ( x  =  B  \/  x  =  C  \/  x  =  A ) }
52, 3, 43eqtr4i 2466 1  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
Colors of variables: wff set class
Syntax hints:    \/ w3o 935    = wceq 1652   {cab 2422   {ctp 3816
This theorem is referenced by:  tpcomb  3901  tpass  3902  tpidm13  3906  tpidm23  3907  tpprceq3  3938  fvtp2  5940  fvtp3  5941  fvtp2g  5943  fvtp3g  5944  en3lplem2  7671  nb3graprlem2  21461  nb3grapr  21462  nb3grapr2  21463  nb3gra2nb  21464  cusgra3v  21473  f13dfv  28081  frgra3v  28392  1to3vfriswmgra  28397  en3lplem2VD  28956  dvh4dimN  32245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821  df-tp 3822
  Copyright terms: Public domain W3C validator