Theorem tpss 4195

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
tpss.2
tpss.3

Assertion

Ref	Expression
tpss

Proof of Theorem tpss

Step	Hyp	Ref	Expression
1		unss 3677	. 2
2		df-3an 975	. . 3
3		tpss.1	. . . . 5
4		tpss.2	. . . . 5
5	3, 4	prss 4184	. . . 4
6		tpss.3	. . . . 5
7	6	snss 4154	. . . 4
8	5, 7	anbi12i 697	. . 3
9	2, 8	bitri 249	. 2
10		df-tp 4034	. . 3
11	10	sseq1i 3527	. 2
12	1, 9, 11	3bitr4i 277	1

Syntax hints:  <->wb 184  /\wa 369  /\w3a 973  e.wcel 1818  cvv 3109  u.cun 3473  C_wss 3475  {csn 4029  {cpr 4031  {ctp 4033

This theorem is referenced by:  1cubr  23173  constr3trllem1  24650  rabren3dioph  30749  fourierdlem102  31991  fourierdlem114  32003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482  df-ss 3489  df-sn 4030  df-pr 4032  df-tp 4034