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Theorem prss 3888
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 3457 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
2 prss.1 . . . 4  |-  A  e. 
_V
32snss 3862 . . 3  |-  ( A  e.  C  <->  { A }  C_  C )
4 prss.2 . . . 4  |-  B  e. 
_V
54snss 3862 . . 3  |-  ( B  e.  C  <->  { B }  C_  C )
63, 5anbi12i 679 . 2  |-  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C ) )
7 df-pr 3757 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
87sseq1i 3308 . 2  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
91, 6, 83bitr4i 269 1  |-  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1717   _Vcvv 2892    u. cun 3254    C_ wss 3256   {csn 3750   {cpr 3751
This theorem is referenced by:  tpss  3900  prsspw  3906  uniintsn  4022  pwssun  4423  xpsspwOLD  4920  dffv2  5728  fiint  7312  wunex2  8539  hashfun  11620  prdsle  13604  prdsless  13605  prdsleval  13619  pwsle  13634  acsfn2  13808  clatl  14463  ipoval  14500  ipolerval  14502  eqgfval  14908  eqgval  14909  gaorb  15004  efgcpbllema  15306  frgpuplem  15324  drngnidl  16220  drnglpir  16244  ltbval  16452  ltbwe  16453  opsrle  16456  opsrtoslem1  16464  thlle  16840  isphtpc  18883  usgrarnedg  21263  cusgrarn  21327  constr2trl  21439  shincli  22705  chincli  22803  coinfliprv  24512  altxpsspw  25529  axlowdimlem4  25591  frgraun  27742  frisusgranb  27743  frgra2v  27745  frgra3vlem1  27746  frgra3vlem2  27747  2pthfrgrarn  27755  frgrancvvdeqlem3  27777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-un 3261  df-in 3263  df-ss 3270  df-sn 3756  df-pr 3757
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