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Theorem tpprceq3 4138
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 491 . 2  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  <-> 
( -.  C  e. 
_V  \/  -.  C  =/=  B ) )
2 tprot 4093 . . . 4  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 df-tp 4002 . . . . 5  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
4 prprc2 4109 . . . . . . 7  |-  ( -.  C  e.  _V  ->  { B ,  C }  =  { B } )
54uneq1d 3620 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  ( { B }  u.  { A } ) )
6 df-pr 4000 . . . . . . 7  |-  { B ,  A }  =  ( { B }  u.  { A } )
7 prcom 4076 . . . . . . 7  |-  { B ,  A }  =  { A ,  B }
86, 7eqtr3i 2454 . . . . . 6  |-  ( { B }  u.  { A } )  =  { A ,  B }
95, 8syl6eq 2480 . . . . 5  |-  ( -.  C  e.  _V  ->  ( { B ,  C }  u.  { A } )  =  { A ,  B }
)
103, 9syl5eq 2476 . . . 4  |-  ( -.  C  e.  _V  ->  { B ,  C ,  A }  =  { A ,  B }
)
112, 10syl5eq 2476 . . 3  |-  ( -.  C  e.  _V  ->  { A ,  B ,  C }  =  { A ,  B }
)
12 nne 2625 . . . 4  |-  ( -.  C  =/=  B  <->  C  =  B )
13 tppreq3 4103 . . . . 5  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  B } )
1413eqcoms 2435 . . . 4  |-  ( C  =  B  ->  { A ,  B ,  C }  =  { A ,  B } )
1512, 14sylbi 199 . . 3  |-  ( -.  C  =/=  B  ->  { A ,  B ,  C }  =  { A ,  B }
)
1611, 15jaoi 381 . 2  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  B
)  ->  { A ,  B ,  C }  =  { A ,  B } )
171, 16sylbi 199 1  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869    =/= wne 2619   _Vcvv 3082    u. cun 3435   {csn 3997   {cpr 3999   {ctp 4001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-v 3084  df-dif 3440  df-un 3442  df-nul 3763  df-sn 3998  df-pr 4000  df-tp 4002
This theorem is referenced by:  tppreqb  4139  1to3vfriswmgra  25727
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