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Theorem tpssg 24932
 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
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpssg.1 . 2
2 tpssg.2 . 2
3 tpssg.3 . 2
4 eleq1 2343 . . . . 5
543anbi1d 1256 . . . 4
6 tpeq1 3715 . . . . 5
76sseq1d 3205 . . . 4
85, 7bibi12d 312 . . 3
9 eleq1 2343 . . . . 5
1093anbi2d 1257 . . . 4
11 tpeq2 3716 . . . . 5
1211sseq1d 3205 . . . 4
1310, 12bibi12d 312 . . 3
14 eleq1 2343 . . . . 5
15143anbi3d 1258 . . . 4
16 tpeq3 3717 . . . . 5
1716sseq1d 3205 . . . 4
1815, 17bibi12d 312 . . 3
19 vex 2791 . . . . 5
20 vex 2791 . . . . 5
21 vex 2791 . . . . 5
2219, 20, 21tpss 3779 . . . 4
2322a1i 10 . . 3
248, 13, 18, 23vtocl3ga 2853 . 2
251, 2, 3, 24syl3anc 1182 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   w3a 934   wceq 1623   wcel 1684   wss 3152  ctp 3642 This theorem is referenced by:  isibg1a6  26125 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-pr 3647  df-tp 3648
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