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Theorem tpss 3655
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 3220 . 2  |-  ( ( { A ,  B }  C_  D  /\  { C }  C_  D )  <-> 
( { A ,  B }  u.  { C } )  C_  D
)
2 df-3an 949 . . 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 3646 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  <->  { A ,  B }  C_  D )
6 tpss.3 . . . . 5  |-  C  e. 
_V
76snss 3619 . . . 4  |-  ( C  e.  D  <->  { C }  C_  D )
85, 7anbi12i 455 . . 3  |-  ( ( ( A  e.  D  /\  B  e.  D
)  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
92, 8bitri 183 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
10 df-tp 3505 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110sseq1i 3093 . 2  |-  ( { A ,  B ,  C }  C_  D  <->  ( { A ,  B }  u.  { C } ) 
C_  D )
121, 9, 113bitr4i 211 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 947    e. wcel 1465   _Vcvv 2660    u. cun 3039    C_ wss 3041   {csn 3497   {cpr 3498   {ctp 3499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-tp 3505
This theorem is referenced by: (None)
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