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Theorem tpssi 3724
Description: A triple of elements of a class is a subset of the class. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Assertion
Ref Expression
tpssi  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B ,  C }  C_  D
)

Proof of Theorem tpssi
StepHypRef Expression
1 df-tp 3569 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
2 prssi 3716 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  ->  { A ,  B }  C_  D )
323adant3 1002 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B }  C_  D )
4 snssi 3702 . . . 4  |-  ( C  e.  D  ->  { C }  C_  D )
543ad2ant3 1005 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { C }  C_  D )
63, 5unssd 3284 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( { A ,  B }  u.  { C } )  C_  D
)
71, 6eqsstrid 3174 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B ,  C }  C_  D
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963    e. wcel 2128    u. cun 3100    C_ wss 3102   {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-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-3an 965  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-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-tp 3569
This theorem is referenced by: (None)
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