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Theorem tpss 3799
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 3347 . 2  |-  ( ( { A ,  B }  C_  D  /\  { C }  C_  D )  <-> 
( { A ,  B }  u.  { C } )  C_  D
)
2 df-3an 983 . . 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 3789 . . . 4  |-  ( ( A  e.  D  /\  B  e.  D )  <->  { A ,  B }  C_  D )
6 tpss.3 . . . . 5  |-  C  e. 
_V
76snss 3768 . . . 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 3641 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
1110sseq1i 3219 . 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 981    e. wcel 2176   _Vcvv 2772    u. cun 3164    C_ wss 3166   {csn 3633   {cpr 3634   {ctp 3635
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-tp 3641
This theorem is referenced by: (None)
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