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Theorem tpprceq3 4114
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 4069 . . . 4  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 df-tp 3979 . . . . 5  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
4 prprc2 4085 . . . . . . 7  |-  ( -.  C  e.  _V  ->  { B ,  C }  =  { B } )
54uneq1d 3598 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  ( { B }  u.  { A } ) )
6 df-pr 3977 . . . . . . 7  |-  { B ,  A }  =  ( { B }  u.  { A } )
7 prcom 4052 . . . . . . 7  |-  { B ,  A }  =  { A ,  B }
86, 7eqtr3i 2435 . . . . . 6  |-  ( { B }  u.  { A } )  =  { A ,  B }
95, 8syl6eq 2461 . . . . 5  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  { A ,  B }
)
103, 9syl5eq 2457 . . . 4  |-  ( -.  C  e.  _V  ->  { B ,  C ,  A }  =  { A ,  B }
)
112, 10syl5eq 2457 . . 3  |-  ( -.  C  e.  _V  ->  { A ,  B ,  C }  =  { A ,  B }
)
12 nne 2606 . . . 4  |-  ( -.  C  =/=  B  <->  C  =  B )
13 tppreq3 4079 . . . . 5  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
1413eqcoms 2416 . . . 4  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B } )
1512, 14sylbi 197 . . 3  |-  ( -.  C  =/=  B  ->  { A ,  B ,  C }  =  { A ,  B }
)
1611, 15jaoi 379 . 2  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  B
)  ->  { A ,  B ,  C }  =  { A ,  B } )
171, 16sylbi 197 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 1407    e. wcel 1844    =/= wne 2600   _Vcvv 3061    u. cun 3414   {csn 3974   {cpr 3976   {ctp 3978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-v 3063  df-dif 3419  df-un 3421  df-nul 3741  df-sn 3975  df-pr 3977  df-tp 3979
This theorem is referenced by:  tppreqb  4115  1to3vfriswmgra  25436
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