Theorem tpss 3835

Description: A triplet of elements of a class is a subset of the class. (Contributed by NM, 9-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypotheses

Ref	Expression
tpss.1	|- A e. _V
tpss.2	|- B e. _V
tpss.3	|- C e. _V

Assertion

Ref	Expression
tpss	|- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D )

Proof of Theorem tpss

Step	Hyp	Ref	Expression
1		unss 3378	. 2 |- ( ( { A , B } C_ D /\ { C } C_ D ) <-> ( { A , B } u. { C } ) C_ D )
2		df-3an 1004	. . 3 |- ( ( A e. D /\ B e. D /\ C e. D ) <-> ( ( A e. D /\ B e. D ) /\ C e. D ) )
3		tpss.1	. . . . 5 |- A e. _V
4		tpss.2	. . . . 5 |- B e. _V
5	3, 4	prss 3823	. . . 4 |- ( ( A e. D /\ B e. D ) <-> { A , B } C_ D )
6		tpss.3	. . . . 5 |- C e. _V
7	6	snss 3802	. . . 4 |- ( C e. D <-> { C } C_ D )
8	5, 7	anbi12i 460	. . 3 |- ( ( ( A e. D /\ B e. D ) /\ C e. D ) <-> ( { A , B } C_ D /\ { C } C_ D ) )
9	2, 8	bitri 184	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) <-> ( { A , B } C_ D /\ { C } C_ D ) )
10		df-tp 3674	. . 3 |- { A , B , C } = ( { A , B } u. { C } )
11	10	sseq1i 3250	. 2 |- ( { A , B , C } C_ D <-> ( { A , B } u. { C } ) C_ D )
12	1, 9, 11	3bitr4i 212	1 |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1002   e. wcel 2200   _Vcvv 2799   u. cun 3195   C_ wss 3197   {csn 3666   {cpr 3667   {ctp 3668

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-tp 3674

This theorem is referenced by: (None)