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Theorem sstpr 3722
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 3721 . . 3  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )
2 prsstp12 3711 . . 3  |-  { B ,  C }  C_  { B ,  C ,  D }
31, 2sstrdi 3140 . 2  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C ,  D } )
4 snsstp3 3710 . . . . 5  |-  { D }  C_  { B ,  C ,  D }
5 sseq1 3151 . . . . 5  |-  ( A  =  { D }  ->  ( A  C_  { B ,  C ,  D }  <->  { D }  C_  { B ,  C ,  D }
) )
64, 5mpbiri 167 . . . 4  |-  ( A  =  { D }  ->  A  C_  { B ,  C ,  D }
)
7 prsstp13 3712 . . . . 5  |-  { B ,  D }  C_  { B ,  C ,  D }
8 sseq1 3151 . . . . 5  |-  ( A  =  { B ,  D }  ->  ( A 
C_  { B ,  C ,  D }  <->  { B ,  D }  C_ 
{ B ,  C ,  D } ) )
97, 8mpbiri 167 . . . 4  |-  ( A  =  { B ,  D }  ->  A  C_  { B ,  C ,  D } )
106, 9jaoi 706 . . 3  |-  ( ( A  =  { D }  \/  A  =  { B ,  D }
)  ->  A  C_  { B ,  C ,  D }
)
11 prsstp23 3713 . . . . 5  |-  { C ,  D }  C_  { B ,  C ,  D }
12 sseq1 3151 . . . . 5  |-  ( A  =  { C ,  D }  ->  ( A 
C_  { B ,  C ,  D }  <->  { C ,  D }  C_ 
{ B ,  C ,  D } ) )
1311, 12mpbiri 167 . . . 4  |-  ( A  =  { C ,  D }  ->  A  C_  { B ,  C ,  D } )
14 eqimss 3182 . . . 4  |-  ( A  =  { B ,  C ,  D }  ->  A  C_  { B ,  C ,  D }
)
1513, 14jaoi 706 . . 3  |-  ( ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } )  ->  A  C_  { B ,  C ,  D }
)
1610, 15jaoi 706 . 2  |-  ( ( ( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )  ->  A  C_  { B ,  C ,  D }
)
173, 16jaoi 706 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 698    = wceq 1335    C_ wss 3102   (/)c0 3395   {csn 3561   {cpr 3562   {ctp 3563
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-sn 3567  df-pr 3568  df-tp 3569
This theorem is referenced by:  pwtpss  3771
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