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Theorem prss 3952
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 3521 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
2 prss.1 . . . 4  |-  A  e. 
_V
32snss 3926 . . 3  |-  ( A  e.  C  <->  { A }  C_  C )
4 prss.2 . . . 4  |-  B  e. 
_V
54snss 3926 . . 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 3821 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
87sseq1i 3372 . 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 1725   _Vcvv 2956    u. cun 3318    C_ wss 3320   {csn 3814   {cpr 3815
This theorem is referenced by:  tpss  3964  prsspw  3971  uniintsn  4087  pwssun  4489  xpsspwOLD  4987  dffv2  5796  fiint  7383  wunex2  8613  hashfun  11700  prdsle  13684  prdsless  13685  prdsleval  13699  pwsle  13714  acsfn2  13888  clatl  14543  ipoval  14580  ipolerval  14582  eqgfval  14988  eqgval  14989  gaorb  15084  efgcpbllema  15386  frgpuplem  15404  drngnidl  16300  drnglpir  16324  ltbval  16532  ltbwe  16533  opsrle  16536  opsrtoslem1  16544  thlle  16924  isphtpc  19019  usgrarnedg  21404  cusgrarn  21468  shincli  22864  chincli  22962  coinfliprv  24740  altxpsspw  25822  axlowdimlem4  25884  frgraun  28386  frisusgranb  28387  frgra2v  28389  frgra3vlem1  28390  frgra3vlem2  28391  2pthfrgrarn  28399  frgrancvvdeqlem3  28421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334  df-sn 3820  df-pr 3821
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