MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpprceq3 Structured version   Unicode version

Theorem tpprceq3 4173
Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
tpprceq3  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B } )

Proof of Theorem tpprceq3
StepHypRef Expression
1 ianor 488 . 2  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  <-> 
( -.  C  e. 
_V  \/  -.  C  =/=  B ) )
2 tprot 4128 . . . 4  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 df-tp 4038 . . . . 5  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
4 prprc2 4144 . . . . . . 7  |-  ( -.  C  e.  _V  ->  { B ,  C }  =  { B } )
54uneq1d 3662 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  ( { B }  u.  { A } ) )
6 df-pr 4036 . . . . . . 7  |-  { B ,  A }  =  ( { B }  u.  { A } )
7 prcom 4111 . . . . . . 7  |-  { B ,  A }  =  { A ,  B }
86, 7eqtr3i 2498 . . . . . 6  |-  ( { B }  u.  { A } )  =  { A ,  B }
95, 8syl6eq 2524 . . . . 5  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  { A ,  B }
)
103, 9syl5eq 2520 . . . 4  |-  ( -.  C  e.  _V  ->  { B ,  C ,  A }  =  { A ,  B }
)
112, 10syl5eq 2520 . . 3  |-  ( -.  C  e.  _V  ->  { A ,  B ,  C }  =  { A ,  B }
)
12 nne 2668 . . . 4  |-  ( -.  C  =/=  B  <->  C  =  B )
13 tppreq3 4138 . . . . 5  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
1413eqcoms 2479 . . . 4  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B } )
1512, 14sylbi 195 . . 3  |-  ( -.  C  =/=  B  ->  { A ,  B ,  C }  =  { A ,  B }
)
1611, 15jaoi 379 . 2  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  B
)  ->  { A ,  B ,  C }  =  { A ,  B } )
171, 16sylbi 195 1  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118    u. cun 3479   {csn 4033   {cpr 4035   {ctp 4037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-nul 3791  df-sn 4034  df-pr 4036  df-tp 4038
This theorem is referenced by:  tppreqb  4174  1to3vfriswmgra  24830
  Copyright terms: Public domain W3C validator