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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
StepHypRef Expression
1 unss 3677 . 2
2 df-3an 975 . . 3
3 tpss.1 . . . . 5
4 tpss.2 . . . . 5
53, 4prss 4184 . . . 4
6 tpss.3 . . . . 5
76snss 4154 . . . 4
85, 7anbi12i 697 . . 3
92, 8bitri 249 . 2
10 df-tp 4034 . . 3
1110sseq1i 3527 . 2
121, 9, 113bitr4i 277 1
Colors of variables: wff setvar class
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
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