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Theorem sstpr 3652
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 3651 . . 3  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )
2 prsstp12 3641 . . 3  |-  { B ,  C }  C_  { B ,  C ,  D }
31, 2syl6ss 3077 . 2  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C ,  D } )
4 snsstp3 3640 . . . . 5  |-  { D }  C_  { B ,  C ,  D }
5 sseq1 3088 . . . . 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 3642 . . . . 5  |-  { B ,  D }  C_  { B ,  C ,  D }
8 sseq1 3088 . . . . 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 688 . . 3  |-  ( ( A  =  { D }  \/  A  =  { B ,  D }
)  ->  A  C_  { B ,  C ,  D }
)
11 prsstp23 3643 . . . . 5  |-  { C ,  D }  C_  { B ,  C ,  D }
12 sseq1 3088 . . . . 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 3119 . . . 4  |-  ( A  =  { B ,  C ,  D }  ->  A  C_  { B ,  C ,  D }
)
1513, 14jaoi 688 . . 3  |-  ( ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } )  ->  A  C_  { B ,  C ,  D }
)
1610, 15jaoi 688 . 2  |-  ( ( ( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )  ->  A  C_  { B ,  C ,  D }
)
173, 16jaoi 688 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 680    = wceq 1314    C_ wss 3039   (/)c0 3331   {csn 3495   {cpr 3496   {ctp 3497
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3or 946  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-tp 3503
This theorem is referenced by:  pwtpss  3701
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