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Theorem tpssi 3598
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 3449 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
2 prssi 3590 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  ->  { A ,  B }  C_  D )
323adant3 963 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B }  C_  D )
4 snssi 3576 . . . 4  |-  ( C  e.  D  ->  { C }  C_  D )
543ad2ant3 966 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { C }  C_  D )
63, 5unssd 3174 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( { A ,  B }  u.  { C } )  C_  D
)
71, 6syl5eqss 3068 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 924    e. wcel 1438    u. cun 2995    C_ wss 2997   {csn 3441   {cpr 3442   {ctp 3443
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-tp 3449
This theorem is referenced by: (None)
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