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Theorem tpss 3795
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 3362 . 2  |-  ( ( { A ,  B }  C_  D  /\  { C }  C_  D )  <-> 
( { A ,  B }  u.  { C } )  C_  D
)
2 df-3an 936 . . 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 3785 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  <->  { A ,  B }  C_  D )
6 tpss.3 . . . . 5  |-  C  e. 
_V
76snss 3761 . . . 4  |-  ( C  e.  D  <->  { C }  C_  D )
85, 7anbi12i 678 . . 3  |-  ( ( ( A  e.  D  /\  B  e.  D
)  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
92, 8bitri 240 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
10 df-tp 3661 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110sseq1i 3215 . 2  |-  ( { A ,  B ,  C }  C_  D  <->  ( { A ,  B }  u.  { C } ) 
C_  D )
121, 9, 113bitr4i 268 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  1cubr  20154  tpssg  25035  rabren3dioph  27001  constr3trllem1  28396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-pr 3660  df-tp 3661
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