Theorem tpssi 4369

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

Step	Hyp	Ref	Expression
1		df-tp 4182	. 2 |- { A , B , C } = ( { A , B } u. { C } )
2		prssi 4353	. . . 4 |- ( ( A e. D /\ B e. D ) -> { A , B } C_ D )
3	2	3adant3 1081	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B } C_ D )
4		snssi 4339	. . . 4 |- ( C e. D -> { C } C_ D )
5	4	3ad2ant3 1084	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { C } C_ D )
6	3, 5	unssd 3789	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( { A , B } u. { C } ) C_ D )
7	1, 6	syl5eqss 3649	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  lcmftp  15349  trgcgrg  25410  sgnclre  30601  signstf  30643  limsupequzlem  39954  fourierdlem46  40369  fourierdlem102  40425  fourierdlem114  40437  etransclem48  40499