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Theorem prss 3785
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypotheses
Ref Expression
prss.1  |-  A  e. 
_V
prss.2  |-  B  e. 
_V
Assertion
Ref Expression
prss  |-  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C )

Proof of Theorem prss
StepHypRef Expression
1 unss 3362 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
2 prss.1 . . . 4  |-  A  e. 
_V
32snss 3761 . . 3  |-  ( A  e.  C  <->  { A }  C_  C )
4 prss.2 . . . 4  |-  B  e. 
_V
54snss 3761 . . 3  |-  ( B  e.  C  <->  { B }  C_  C )
63, 5anbi12i 678 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C ) )
7 df-pr 3660 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
87sseq1i 3215 . 2  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
91, 6, 83bitr4i 268 1  |-  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653   {cpr 3654
This theorem is referenced by:  tpss  3795  prsspw  3801  uniintsn  3915  pwssun  4315  xpsspwOLD  4814  dffv2  5608  fiint  7149  wunex2  8376  hashfun  11405  prdsle  13377  prdsless  13378  prdsleval  13392  pwsle  13407  acsfn2  13581  clatl  14236  ipoval  14273  ipolerval  14275  eqgfval  14681  eqgval  14682  gaorb  14777  efgcpbllema  15079  frgpuplem  15097  drngnidl  15997  drnglpir  16021  ltbval  16229  ltbwe  16230  opsrle  16233  opsrtoslem1  16241  thlle  16613  isphtpc  18508  shincli  21957  chincli  22055  altxpsspw  24583  axlowdimlem4  24645  toplat  25393  pgapspf2  26156  frgra2v  28423  frgra3vlem1  28424  frgra3vlem2  28425  2pthfrgrarn  28433
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-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
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