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

Theorem tpcoma 3723
Description: Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)
Assertion
Ref Expression
tpcoma  |-  { A ,  B ,  C }  =  { B ,  A ,  C }

Proof of Theorem tpcoma
StepHypRef Expression
1 prcom 3705 . . 3  |-  { A ,  B }  =  { B ,  A }
21uneq1i 3325 . 2  |-  ( { A ,  B }  u.  { C } )  =  ( { B ,  A }  u.  { C } )
3 df-tp 3648 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
4 df-tp 3648 . 2  |-  { B ,  A ,  C }  =  ( { B ,  A }  u.  { C } )
52, 3, 43eqtr4i 2313 1  |-  { A ,  B ,  C }  =  { B ,  A ,  C }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150   {csn 3640   {cpr 3641   {ctp 3642
This theorem is referenced by:  tpcomb  3724  frgra3v  28180  3vfriswmgra  28183  1to3vfriswmgra  28185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-pr 3647  df-tp 3648
  Copyright terms: Public domain W3C validator