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Theorem tpss 3956
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  |-  A  e. 
_V
tpss.2  |-  B  e. 
_V
tpss.3  |-  C  e. 
_V
Assertion
Ref Expression
tpss  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D )

Proof of Theorem tpss
StepHypRef Expression
1 unss 3513 . 2  |-  ( ( { A ,  B }  C_  D  /\  { C }  C_  D )  <-> 
( { A ,  B }  u.  { C } )  C_  D
)
2 df-3an 938 . . 3  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  ( ( A  e.  D  /\  B  e.  D
)  /\  C  e.  D ) )
3 tpss.1 . . . . 5  |-  A  e. 
_V
4 tpss.2 . . . . 5  |-  B  e. 
_V
53, 4prss 3944 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  <->  { A ,  B }  C_  D )
6 tpss.3 . . . . 5  |-  C  e. 
_V
76snss 3918 . . . 4  |-  ( C  e.  D  <->  { C }  C_  D )
85, 7anbi12i 679 . . 3  |-  ( ( ( A  e.  D  /\  B  e.  D
)  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
92, 8bitri 241 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
10 df-tp 3814 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110sseq1i 3364 . 2  |-  ( { A ,  B ,  C }  C_  D  <->  ( { A ,  B }  u.  { C } ) 
C_  D )
121, 9, 113bitr4i 269 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   _Vcvv 2948    u. cun 3310    C_ wss 3312   {csn 3806   {cpr 3807   {ctp 3808
This theorem is referenced by:  1cubr  20672  constr3trllem1  21627  rabren3dioph  26830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-sn 3812  df-pr 3813  df-tp 3814
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