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Theorem prssg 3945
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 3924 . . 3  |-  ( A  e.  V  ->  ( A  e.  C  <->  { A }  C_  C ) )
2 snssg 3924 . . 3  |-  ( B  e.  W  ->  ( B  e.  C  <->  { B }  C_  C ) )
31, 2bi2anan9 844 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C
) ) )
4 unss 3513 . . 3  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
5 df-pr 3813 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
65sseq1i 3364 . . 3  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
74, 6bitr4i 244 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  { A ,  B }  C_  C
)
83, 7syl6bb 253 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 4    <-> wb 177    /\ wa 359    e. wcel 1725    u. cun 3310    C_ wss 3312   {csn 3806   {cpr 3807
This theorem is referenced by:  prssi  3946  prsspwg  3947  prsspwgOLD  3948  lspprss  16060  lspvadd  16160  topgele  16991  usgraedgprv  21388  usgraedgrnv  21389  usgraedg4  21398  2trllemH  21544  2trllemE  21545  dihmeetlem2N  32034
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-sn 3812  df-pr 3813
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