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Theorem diftpsneq 28081
Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
diftpsneq  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )

Proof of Theorem diftpsneq
StepHypRef Expression
1 df-tp 3650 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
21a1i 10 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } ) )
32difeq1d 3295 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  ( ( { A ,  B }  u.  { C } ) 
\  { C }
) )
4 difundir 3424 . . 3  |-  ( ( { A ,  B }  u.  { C } )  \  { C } )  =  ( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )
54a1i 10 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  u.  { C } )  \  { C } )  =  ( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) ) )
6 df-pr 3649 . . . . . . . . 9  |-  { A ,  B }  =  ( { A }  u.  { B } )
76a1i 10 . . . . . . . 8  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B }  =  ( { A }  u.  { B } ) )
87ineq1d 3371 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C }
) )
9 incom 3363 . . . . . . . . 9  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( { C }  i^i  ( { A }  u.  { B } ) )
10 indi 3417 . . . . . . . . 9  |-  ( { C }  i^i  ( { A }  u.  { B } ) )  =  ( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )
119, 10eqtri 2305 . . . . . . . 8  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( ( { C }  i^i  { A } )  u.  ( { C }  i^i  { B } ) )
1211a1i 10 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A }  u.  { B } )  i^i  { C } )  =  ( ( { C }  i^i  { A } )  u.  ( { C }  i^i  { B }
) ) )
13 necom 2529 . . . . . . . . . . 11  |-  ( A  =/=  C  <->  C  =/=  A )
14 disjsn2 3696 . . . . . . . . . . 11  |-  ( C  =/=  A  ->  ( { C }  i^i  { A } )  =  (/) )
1513, 14sylbi 187 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { C }  i^i  { A } )  =  (/) )
1615adantr 451 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  i^i  { A } )  =  (/) )
17 necom 2529 . . . . . . . . . . 11  |-  ( B  =/=  C  <->  C  =/=  B )
18 disjsn2 3696 . . . . . . . . . . 11  |-  ( C  =/=  B  ->  ( { C }  i^i  { B } )  =  (/) )
1917, 18sylbi 187 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { C }  i^i  { B } )  =  (/) )
2019adantl 452 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  i^i  { B } )  =  (/) )
2116, 20uneq12d 3332 . . . . . . . 8  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )  =  ( (/)  u.  (/) ) )
22 unidm 3320 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
2321, 22syl6eq 2333 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )  =  (/) )
248, 12, 233eqtrd 2321 . . . . . 6  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
25 disj3 3501 . . . . . 6  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  { A ,  B }  =  ( { A ,  B }  \  { C }
) )
2624, 25sylib 188 . . . . 5  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B }  =  ( { A ,  B }  \  { C } ) )
2726eqcomd 2290 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  \  { C } )  =  { A ,  B }
)
28 difid 3524 . . . . 5  |-  ( { C }  \  { C } )  =  (/)
2928a1i 10 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  \  { C } )  =  (/) )
3027, 29uneq12d 3332 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )  =  ( { A ,  B }  u.  (/) ) )
31 un0 3481 . . 3  |-  ( { A ,  B }  u.  (/) )  =  { A ,  B }
3230, 31syl6eq 2333 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )  =  { A ,  B } )
333, 5, 323eqtrd 2321 1  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    =/= wne 2448    \ cdif 3151    u. cun 3152    i^i cin 3153   (/)c0 3457   {csn 3642   {cpr 3643   {ctp 3644
This theorem is referenced by:  frgra3v  28191  3vfriswmgra  28194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-sn 3648  df-pr 3649  df-tp 3650
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