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Theorem diftpsn3 3669
Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
diftpsn3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )

Proof of Theorem diftpsn3
StepHypRef Expression
1 df-tp 3540 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
21a1i 9 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } ) )
32difeq1d 3198 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  ( ( { A ,  B }  u.  { C } ) 
\  { C }
) )
4 difundir 3334 . . 3  |-  ( ( { A ,  B }  u.  { C } )  \  { C } )  =  ( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )
54a1i 9 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  u.  { C } )  \  { C } )  =  ( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) ) )
6 df-pr 3539 . . . . . . . . 9  |-  { A ,  B }  =  ( { A }  u.  { B } )
76a1i 9 . . . . . . . 8  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B }  =  ( { A }  u.  { B } ) )
87ineq1d 3281 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C }
) )
9 incom 3273 . . . . . . . . 9  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( { C }  i^i  ( { A }  u.  { B } ) )
10 indi 3328 . . . . . . . . 9  |-  ( { C }  i^i  ( { A }  u.  { B } ) )  =  ( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )
119, 10eqtri 2161 . . . . . . . 8  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( ( { C }  i^i  { A } )  u.  ( { C }  i^i  { B } ) )
1211a1i 9 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A }  u.  { B } )  i^i  { C } )  =  ( ( { C }  i^i  { A } )  u.  ( { C }  i^i  { B }
) ) )
13 necom 2393 . . . . . . . . . . 11  |-  ( A  =/=  C  <->  C  =/=  A )
14 disjsn2 3594 . . . . . . . . . . 11  |-  ( C  =/=  A  ->  ( { C }  i^i  { A } )  =  (/) )
1513, 14sylbi 120 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { C }  i^i  { A } )  =  (/) )
1615adantr 274 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  i^i  { A } )  =  (/) )
17 necom 2393 . . . . . . . . . . 11  |-  ( B  =/=  C  <->  C  =/=  B )
18 disjsn2 3594 . . . . . . . . . . 11  |-  ( C  =/=  B  ->  ( { C }  i^i  { B } )  =  (/) )
1917, 18sylbi 120 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { C }  i^i  { B } )  =  (/) )
2019adantl 275 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  i^i  { B } )  =  (/) )
2116, 20uneq12d 3236 . . . . . . . 8  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )  =  ( (/)  u.  (/) ) )
22 unidm 3224 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
2321, 22eqtrdi 2189 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )  =  (/) )
248, 12, 233eqtrd 2177 . . . . . 6  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
25 disj3 3420 . . . . . 6  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  { A ,  B }  =  ( { A ,  B }  \  { C }
) )
2624, 25sylib 121 . . . . 5  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B }  =  ( { A ,  B }  \  { C } ) )
2726eqcomd 2146 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  \  { C } )  =  { A ,  B }
)
28 difid 3436 . . . . 5  |-  ( { C }  \  { C } )  =  (/)
2928a1i 9 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  \  { C } )  =  (/) )
3027, 29uneq12d 3236 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )  =  ( { A ,  B }  u.  (/) ) )
31 un0 3401 . . 3  |-  ( { A ,  B }  u.  (/) )  =  { A ,  B }
3230, 31eqtrdi 2189 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )  =  { A ,  B } )
333, 5, 323eqtrd 2177 1  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    =/= wne 2309    \ cdif 3073    u. cun 3074    i^i cin 3075   (/)c0 3368   {csn 3532   {cpr 3533   {ctp 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-tp 3540
This theorem is referenced by: (None)
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