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Theorem tpss 3520
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  _V
tpss.2  _V
tpss.3  C 
_V
Assertion
Ref Expression
tpss  D  D  C  D  { ,  ,  C }  C_  D

Proof of Theorem tpss
StepHypRef Expression
1 unss 3111 . 2  { ,  }  C_  D  { C }  C_  D  { ,  }  u.  { C }  C_  D
2 df-3an 886 . . 3  D  D  C  D  D  D  C  D
3 tpss.1 . . . . 5  _V
4 tpss.2 . . . . 5  _V
53, 4prss 3511 . . . 4  D  D  { ,  }  C_  D
6 tpss.3 . . . . 5  C 
_V
76snss 3485 . . . 4  C  D  { C }  C_  D
85, 7anbi12i 433 . . 3  D  D  C  D  { ,  }  C_  D  { C }  C_  D
92, 8bitri 173 . 2  D  D  C  D  { ,  }  C_  D  { C }  C_  D
10 df-tp 3375 . . 3  { ,  ,  C }  { ,  }  u.  { C }
1110sseq1i 2963 . 2  { ,  ,  C }  C_  D  { ,  }  u.  { C } 
C_  D
121, 9, 113bitr4i 201 1  D  D  C  D  { ,  ,  C }  C_  D
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   w3a 884   wcel 1390   _Vcvv 2551    u. cun 2909    C_ wss 2911   {csn 3367   {cpr 3368   {ctp 3369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-tp 3375
This theorem is referenced by: (None)
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