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Theorem diftpsn3 3732
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 3599 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
21a1i 9 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } ) )
32difeq1d 3252 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  ( ( { A ,  B }  u.  { C } ) 
\  { C }
) )
4 difundir 3388 . . 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 3598 . . . . . . . . 9  |-  { A ,  B }  =  ( { A }  u.  { B } )
76a1i 9 . . . . . . . 8  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B }  =  ( { A }  u.  { B } ) )
87ineq1d 3335 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C }
) )
9 incom 3327 . . . . . . . . 9  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( { C }  i^i  ( { A }  u.  { B } ) )
10 indi 3382 . . . . . . . . 9  |-  ( { C }  i^i  ( { A }  u.  { B } ) )  =  ( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )
119, 10eqtri 2198 . . . . . . . 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 2431 . . . . . . . . . . 11  |-  ( A  =/=  C  <->  C  =/=  A )
14 disjsn2 3654 . . . . . . . . . . 11  |-  ( C  =/=  A  ->  ( { C }  i^i  { A } )  =  (/) )
1513, 14sylbi 121 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { C }  i^i  { A } )  =  (/) )
1615adantr 276 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  i^i  { A } )  =  (/) )
17 necom 2431 . . . . . . . . . . 11  |-  ( B  =/=  C  <->  C  =/=  B )
18 disjsn2 3654 . . . . . . . . . . 11  |-  ( C  =/=  B  ->  ( { C }  i^i  { B } )  =  (/) )
1917, 18sylbi 121 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { C }  i^i  { B } )  =  (/) )
2019adantl 277 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  i^i  { B } )  =  (/) )
2116, 20uneq12d 3290 . . . . . . . 8  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )  =  ( (/)  u.  (/) ) )
22 unidm 3278 . . . . . . . 8  |-  ( (/)  u.  (/) )  =  (/)
2321, 22eqtrdi 2226 . . . . . . 7  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { C }  i^i  { A }
)  u.  ( { C }  i^i  { B } ) )  =  (/) )
248, 12, 233eqtrd 2214 . . . . . 6  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
25 disj3 3475 . . . . . 6  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  { A ,  B }  =  ( { A ,  B }  \  { C }
) )
2624, 25sylib 122 . . . . 5  |-  ( ( A  =/=  C  /\  B  =/=  C )  ->  { A ,  B }  =  ( { A ,  B }  \  { C } ) )
2726eqcomd 2183 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B }  \  { C } )  =  { A ,  B }
)
28 difid 3491 . . . . 5  |-  ( { C }  \  { C } )  =  (/)
2928a1i 9 . . . 4  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { C }  \  { C } )  =  (/) )
3027, 29uneq12d 3290 . . 3  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )  =  ( { A ,  B }  u.  (/) ) )
31 un0 3456 . . 3  |-  ( { A ,  B }  u.  (/) )  =  { A ,  B }
3230, 31eqtrdi 2226 . 2  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( ( { A ,  B }  \  { C } )  u.  ( { C }  \  { C } ) )  =  { A ,  B } )
333, 5, 323eqtrd 2214 1  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A ,  B ,  C }  \  { C } )  =  { A ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    =/= wne 2347    \ cdif 3126    u. cun 3127    i^i cin 3128   (/)c0 3422   {csn 3591   {cpr 3592   {ctp 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-pr 3598  df-tp 3599
This theorem is referenced by: (None)
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