Theorem tpssi 3800

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 3641	. 2 |- { A , B , C } = ( { A , B } u. { C } )
2		prssi 3791	. . . 4 |- ( ( A e. D /\ B e. D ) -> { A , B } C_ D )
3	2	3adant3 1020	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B } C_ D )
4		snssi 3777	. . . 4 |- ( C e. D -> { C } C_ D )
5	4	3ad2ant3 1023	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { C } C_ D )
6	3, 5	unssd 3349	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) -> ( { A , B } u. { C } ) C_ D )
7	1, 6	eqsstrid 3239	1 |- ( ( A e. D /\ B e. D /\ C e. D ) -> { A , B , C } C_ D )

Syntax hints:   -> wi 4   /\ w3a 981   e. wcel 2176   u. cun 3164   C_ wss 3166   {csn 3633   {cpr 3634   {ctp 3635

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-tp 3641

This theorem is referenced by: (None)