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Theorem tpssi 4130
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 3964 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
2 prssi 4119 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  ->  { A ,  B }  C_  D )
323adant3 1050 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B }  C_  D )
4 snssi 4107 . . . 4  |-  ( C  e.  D  ->  { C }  C_  D )
543ad2ant3 1053 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { C }  C_  D )
63, 5unssd 3601 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( { A ,  B }  u.  { C } )  C_  D
)
71, 6syl5eqss 3462 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B ,  C }  C_  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    e. wcel 1904    u. cun 3388    C_ wss 3390   {csn 3959   {cpr 3961   {ctp 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-in 3397  df-ss 3404  df-sn 3960  df-pr 3962  df-tp 3964
This theorem is referenced by:  lcmftp  14688  trgcgrg  24639  2trllemG  25367  sgnclre  29483  signstf  29527  fourierdlem46  38128  fourierdlem102  38184  fourierdlem114  38196  etransclem48OLD  38259  etransclem48  38260
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