Theorem tpssi 3785

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 3626	. 2 |- { A , B , C } = ( { A , B } u. { C } )
2		prssi 3776	. . . 4 |- ( ( A e. D /\ B e. D ) -> { A , B } C_ D )
3	2	3adant3 1019	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B } C_ D )
4		snssi 3762	. . . 4 |- ( C e. D -> { C } C_ D )
5	4	3ad2ant3 1022	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { C } C_ D )
6	3, 5	unssd 3335	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( { A , B } u. { C } ) C_ D )
7	1, 6	eqsstrid 3225	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )

Syntax hints:   -> wi 4   /\ w3a 980   e. wcel 2164   u. cun 3151   C_ wss 3153   {csn 3618   {cpr 3619   {ctp 3620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-tp 3626

This theorem is referenced by: (None)