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Theorem tpprceq3 4013
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 3970 . . . 4  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 df-tp 3882 . . . . 5  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
4 prprc2 3986 . . . . . . 7  |-  ( -.  C  e.  _V  ->  { B ,  C }  =  { B } )
54uneq1d 3509 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  ( { B }  u.  { A } ) )
6 df-pr 3880 . . . . . . 7  |-  { B ,  A }  =  ( { B }  u.  { A } )
7 prcom 3953 . . . . . . 7  |-  { B ,  A }  =  { A ,  B }
86, 7eqtr3i 2465 . . . . . 6  |-  ( { B }  u.  { A } )  =  { A ,  B }
95, 8syl6eq 2491 . . . . 5  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  { A ,  B }
)
103, 9syl5eq 2487 . . . 4  |-  ( -.  C  e.  _V  ->  { B ,  C ,  A }  =  { A ,  B }
)
112, 10syl5eq 2487 . . 3  |-  ( -.  C  e.  _V  ->  { A ,  B ,  C }  =  { A ,  B }
)
12 nne 2612 . . . 4  |-  ( -.  C  =/=  B  <->  C  =  B )
13 tppreq3 3980 . . . . 5  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
1413eqcoms 2446 . . . 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 1369    e. wcel 1756    =/= wne 2606   _Vcvv 2972    u. cun 3326   {csn 3877   {cpr 3879   {ctp 3881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-v 2974  df-dif 3331  df-un 3333  df-nul 3638  df-sn 3878  df-pr 3880  df-tp 3882
This theorem is referenced by:  tppreqb  4014  1to3vfriswmgra  30599
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