Theorem tpss 3871

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 3437	. 2 |- (({A, B} C_ D /\ {C} C_ D) <-> ({A, B} u. {C}) C_ D)
2		df-3an 936	. . 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 3861	. . . 4 |- ((A e. D /\ B e. D) <-> {A, B} C_ D)
6		tpss.3	. . . . 5 |- C e. _V
7	6	snss 3838	. . . 4 |- (C e. D <-> {C} C_ D)
8	5, 7	anbi12i 678	. . 3 |- (((A e. D /\ B e. D) /\ C e. D) <-> ({A, B} C_ D /\ {C} C_ D))
9	2, 8	bitri 240	. 2 |- ((A e. D /\ B e. D /\ C e. D) <-> ({A, B} C_ D /\ {C} C_ D))
10		df-tp 3743	. . 3 |- {A, B, C} = ({A, B} u. {C})
11	10	sseq1i 3295	. 2 |- ({A, B, C} C_ D <-> ({A, B} u. {C}) C_ D)
12	1, 9, 11	3bitr4i 268	1 |- ((A e. D /\ B e. D /\ C e. D) <-> {A, B, C} C_ D)

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934   e. wcel 1710  _Vcvv 2859   u. cun 3207   C_ wss 3257  {csn 3737  {cpr 3738  {ctp 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by: (None)