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Theorem sstpr 3569
Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
sstpr  |-  ( ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  \/  (
( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) )  ->  A  C_  { B ,  C ,  D }
)

Proof of Theorem sstpr
StepHypRef Expression
1 ssprr 3568 . . 3  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )
2 prsstp12 3558 . . 3  |-  { B ,  C }  C_  { B ,  C ,  D }
31, 2syl6ss 3020 . 2  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C ,  D } )
4 snsstp3 3557 . . . . 5  |-  { D }  C_  { B ,  C ,  D }
5 sseq1 3029 . . . . 5  |-  ( A  =  { D }  ->  ( A  C_  { B ,  C ,  D }  <->  { D }  C_  { B ,  C ,  D }
) )
64, 5mpbiri 166 . . . 4  |-  ( A  =  { D }  ->  A  C_  { B ,  C ,  D }
)
7 prsstp13 3559 . . . . 5  |-  { B ,  D }  C_  { B ,  C ,  D }
8 sseq1 3029 . . . . 5  |-  ( A  =  { B ,  D }  ->  ( A 
C_  { B ,  C ,  D }  <->  { B ,  D }  C_ 
{ B ,  C ,  D } ) )
97, 8mpbiri 166 . . . 4  |-  ( A  =  { B ,  D }  ->  A  C_  { B ,  C ,  D } )
106, 9jaoi 669 . . 3  |-  ( ( A  =  { D }  \/  A  =  { B ,  D }
)  ->  A  C_  { B ,  C ,  D }
)
11 prsstp23 3560 . . . . 5  |-  { C ,  D }  C_  { B ,  C ,  D }
12 sseq1 3029 . . . . 5  |-  ( A  =  { C ,  D }  ->  ( A 
C_  { B ,  C ,  D }  <->  { C ,  D }  C_ 
{ B ,  C ,  D } ) )
1311, 12mpbiri 166 . . . 4  |-  ( A  =  { C ,  D }  ->  A  C_  { B ,  C ,  D } )
14 eqimss 3060 . . . 4  |-  ( A  =  { B ,  C ,  D }  ->  A  C_  { B ,  C ,  D }
)
1513, 14jaoi 669 . . 3  |-  ( ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } )  ->  A  C_  { B ,  C ,  D }
)
1610, 15jaoi 669 . 2  |-  ( ( ( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )  ->  A  C_  { B ,  C ,  D }
)
173, 16jaoi 669 1  |-  ( ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  \/  (
( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) )  ->  A  C_  { B ,  C ,  D }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1285    C_ wss 2982   (/)c0 3267   {csn 3416   {cpr 3417   {ctp 3418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3or 921  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-pr 3423  df-tp 3424
This theorem is referenced by:  pwtpss  3618
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