Theorem tpssi 3551

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 3406	. 2 |- { A , B , C } = ( { A , B } u. { C } )
2		prssi 3543	. . . 4 |- ( ( A e. D /\ B e. D ) -> { A , B } C_ D )
3	2	3adant3 958	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B } C_ D )
4		snssi 3529	. . . 4 |- ( C e. D -> { C } C_ D )
5	4	3ad2ant3 961	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { C } C_ D )
6	3, 5	unssd 3148	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( { A , B } u. { C } ) C_ D )
7	1, 6	syl5eqss 3043	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )

Syntax hints:   -> wi 4   /\ w3a 919   e. wcel 1433   u. cun 2971   C_ wss 2973   {csn 3398   {cpr 3399   {ctp 3400

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-tp 3406

This theorem is referenced by: (None)