Theorem tpssi 3842

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 3677	. 2 |- { A , B , C } = ( { A , B } u. { C } )
2		prssi 3831	. . . 4 |- ( ( A e. D /\ B e. D ) -> { A , B } C_ D )
3	2	3adant3 1043	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B } C_ D )
4		snssi 3817	. . . 4 |- ( C e. D -> { C } C_ D )
5	4	3ad2ant3 1046	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { C } C_ D )
6	3, 5	unssd 3383	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( { A , B } u. { C } ) C_ D )
7	1, 6	eqsstrid 3273	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )

Syntax hints:   -> wi 4   /\ w3a 1004   e. wcel 2202   u. cun 3198   C_ wss 3200   {csn 3669   {cpr 3670   {ctp 3671

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by: (None)