Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tpssg Unicode version

Theorem tpssg 24097
 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.) (Revised by FL, 17-May-2016.)
Hypotheses
Ref Expression
tpssg.1
tpssg.2
tpssg.3
Assertion
Ref Expression
tpssg

Proof of Theorem tpssg
StepHypRef Expression
1 tpssg.1 . 2
2 tpssg.2 . 2
3 tpssg.3 . 2
4 eleq1 2313 . . . . 5
543anbi1d 1261 . . . 4
6 tpeq1 3619 . . . . 5
76sseq1d 3126 . . . 4
85, 7bibi12d 314 . . 3
9 eleq1 2313 . . . . 5
1093anbi2d 1262 . . . 4
11 tpeq2 3620 . . . . 5
1211sseq1d 3126 . . . 4
1310, 12bibi12d 314 . . 3
14 eleq1 2313 . . . . 5
15143anbi3d 1263 . . . 4
16 tpeq3 3621 . . . . 5
1716sseq1d 3126 . . . 4
1815, 17bibi12d 314 . . 3
19 vex 2730 . . . . 5
20 vex 2730 . . . . 5
21 vex 2730 . . . . 5
2219, 20, 21tpss 3679 . . . 4
2322a1i 12 . . 3
248, 13, 18, 23vtocl3ga 2791 . 2
251, 2, 3, 24syl3anc 1187 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   w3a 939   wceq 1619   wcel 1621   wss 3078  ctp 3546 This theorem is referenced by:  isibg1a6  25291 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-in 3085  df-ss 3089  df-sn 3550  df-pr 3551  df-tp 3552
 Copyright terms: Public domain W3C validator