Theorem tpss 3770

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 3321	. 2 |- ( ( { A , B } C_ D /\ { C } C_ D ) <-> ( { A , B } u. { C } ) C_ D )
2		df-3an 981	. . 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 3760	. . . 4 |- ( ( A e. D /\ B e. D ) <-> { A , B } C_ D )
6		tpss.3	. . . . 5 |- C e. _V
7	6	snss 3739	. . . 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 3612	. . 3 |- { A , B , C } = ( { A , B } u. { C } )
11	10	sseq1i 3193	. 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 979   e. wcel 2158   _Vcvv 2749   u. cun 3139   C_ wss 3141   {csn 3604   {cpr 3605   {ctp 3606

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-tp 3612

This theorem is referenced by: (None)