MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpssi Structured version   Visualization version   Unicode version

Theorem tpssi 4138
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 3973 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
2 prssi 4128 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  ->  { A ,  B }  C_  D )
323adant3 1028 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { A ,  B }  C_  D )
4 snssi 4116 . . . 4  |-  ( C  e.  D  ->  { C }  C_  D )
543ad2ant3 1031 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  { C }  C_  D )
63, 5unssd 3610 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  ->  ( { A ,  B }  u.  { C } )  C_  D
)
71, 6syl5eqss 3476 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 985    e. wcel 1887    u. cun 3402    C_ wss 3404   {csn 3968   {cpr 3970   {ctp 3972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-un 3409  df-in 3411  df-ss 3418  df-sn 3969  df-pr 3971  df-tp 3973
This theorem is referenced by:  lcmftp  14609  trgcgrg  24560  2trllemG  25288  sgnclre  29410  signstf  29455  fourierdlem46  38016  fourierdlem102  38072  fourierdlem114  38084  etransclem48OLD  38147  etransclem48  38148
  Copyright terms: Public domain W3C validator