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Theorem prssg 3730
Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
prssg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C
) )

Proof of Theorem prssg
StepHypRef Expression
1 snssg 3714 . . 3  |-  ( A  e.  V  ->  ( A  e.  C  <->  { A }  C_  C ) )
2 snssg 3714 . . 3  |-  ( B  e.  W  ->  ( B  e.  C  <->  { B }  C_  C ) )
31, 2bi2anan9 848 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C
) ) )
4 unss 3310 . . 3  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
5 df-pr 3607 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
65sseq1i 3163 . . 3  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
74, 6bitr4i 245 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  { A ,  B }  C_  C
)
83, 7syl6bb 254 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621    u. cun 3111    C_ wss 3113   {csn 3600   {cpr 3601
This theorem is referenced by:  prssi  3731  lspprss  15697  lspvadd  15797  topgele  16620  inttop4  24870  pgapspf2  25406  dihmeetlem2N  30640
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-un 3118  df-in 3120  df-ss 3127  df-sn 3606  df-pr 3607
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