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Theorem tpss 3770
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 3321 . 2  |-  ( ( { A ,  B }  C_  D  /\  { C }  C_  D )  <-> 
( { A ,  B }  u.  { C } )  C_  D
)
2 df-3an 981 . . 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 3760 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  <->  { A ,  B }  C_  D )
6 tpss.3 . . . . 5  |-  C  e. 
_V
76snss 3739 . . . 4  |-  ( C  e.  D  <->  { C }  C_  D )
85, 7anbi12i 460 . . 3  |-  ( ( ( A  e.  D  /\  B  e.  D
)  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
92, 8bitri 184 . 2  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  ( { A ,  B }  C_  D  /\  { C }  C_  D ) )
10 df-tp 3612 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110sseq1i 3193 . 2  |-  ( { A ,  B ,  C }  C_  D  <->  ( { A ,  B }  u.  { C } ) 
C_  D )
121, 9, 113bitr4i 212 1  |-  ( ( A  e.  D  /\  B  e.  D  /\  C  e.  D )  <->  { A ,  B ,  C }  C_  D )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 979    e. wcel 2158   _Vcvv 2749    u. cun 3139    C_ wss 3141   {csn 3604   {cpr 3605   {ctp 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-tp 3612
This theorem is referenced by: (None)
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