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Theorem sstpr 3744
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 3743 . . 3  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )
2 prsstp12 3733 . . 3  |-  { B ,  C }  C_  { B ,  C ,  D }
31, 2sstrdi 3159 . 2  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C ,  D } )
4 snsstp3 3732 . . . . 5  |-  { D }  C_  { B ,  C ,  D }
5 sseq1 3170 . . . . 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 3734 . . . . 5  |-  { B ,  D }  C_  { B ,  C ,  D }
8 sseq1 3170 . . . . 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 711 . . 3  |-  ( ( A  =  { D }  \/  A  =  { B ,  D }
)  ->  A  C_  { B ,  C ,  D }
)
11 prsstp23 3735 . . . . 5  |-  { C ,  D }  C_  { B ,  C ,  D }
12 sseq1 3170 . . . . 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 3201 . . . 4  |-  ( A  =  { B ,  C ,  D }  ->  A  C_  { B ,  C ,  D }
)
1513, 14jaoi 711 . . 3  |-  ( ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } )  ->  A  C_  { B ,  C ,  D }
)
1610, 15jaoi 711 . 2  |-  ( ( ( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )  ->  A  C_  { B ,  C ,  D }
)
173, 16jaoi 711 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 703    = wceq 1348    C_ wss 3121   (/)c0 3414   {csn 3583   {cpr 3584   {ctp 3585
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3or 974  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-tp 3591
This theorem is referenced by:  pwtpss  3793
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