Theorem tpss 3738

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 3296	. 2 |- ( ( { A , B } C_ D /\ { C } C_ D ) <-> ( { A , B } u. { C } ) C_ D )
2		df-3an 970	. . 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 3729	. . . 4 |- ( ( A e. D /\ B e. D ) <-> { A , B } C_ D )
6		tpss.3	. . . . 5 |- C e. _V
7	6	snss 3702	. . . 4 |- ( C e. D <-> { C } C_ D )
8	5, 7	anbi12i 456	. . 3 |- ( ( ( A e. D /\ B e. D ) /\ C e. D ) <-> ( { A , B } C_ D /\ { C } C_ D ) )
9	2, 8	bitri 183	. 2 |- ( ( A e. D /\ B e. D /\ C e. D ) <-> ( { A , B } C_ D /\ { C } C_ D ) )
10		df-tp 3584	. . 3 |- { A , B , C } = ( { A , B } u. { C } )
11	10	sseq1i 3168	. 2 |- ( { A , B , C } C_ D <-> ( { A , B } u. { C } ) C_ D )
12	1, 9, 11	3bitr4i 211	1 |- ( ( A e. D /\ B e. D /\ C e. D ) <-> { A , B , C } C_ D )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 968   e. wcel 2136   _Vcvv 2726   u. cun 3114   C_ wss 3116   {csn 3576   {cpr 3577   {ctp 3578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-tp 3584

This theorem is referenced by: (None)