Theorem tpss 3520

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 3111	. 2 |- (( {A , B } C_ D /\ { C } C_ D) <-> ( {A , B } u. { C }) C_ D)
2		df-3an 886	. . 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 3511	. . . 4 |- ((A e. D /\ B e. D) <-> {A , B } C_ D)
6		tpss.3	. . . . 5 |- C e. _V
7	6	snss 3485	. . . 4 |- ( C e. D <-> { C } C_ D)
8	5, 7	anbi12i 433	. . 3 |- (((A e. D /\ B e. D) /\ C e. D) <-> ( {A , B } C_ D /\ { C } C_ D))
9	2, 8	bitri 173	. 2 |- ((A e. D /\ B e. D /\ C e. D) <-> ( {A , B } C_ D /\ { C } C_ D))
10		df-tp 3375	. . 3 |- {A , B , C } = ( {A , B } u. { C })
11	10	sseq1i 2963	. 2 |- ( {A , B , C } C_ D <-> ( {A , B } u. { C }) C_ D)
12	1, 9, 11	3bitr4i 201	1 |- ((A e. D /\ B e. D /\ C e. D) <-> {A , B , C } C_ D)

Syntax hints:   /\ wa 97   <-> wb 98   /\ w3a 884   e. wcel 1390   _Vcvv 2551   u. cun 2909   C_ wss 2911   {csn 3367   {cpr 3368   {ctp 3369

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-tp 3375

This theorem is referenced by: (None)