MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prssg Unicode version

Theorem prssg 3786
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 3767 . . 3  |-  ( A  e.  V  ->  ( A  e.  C  <->  { A }  C_  C ) )
2 snssg 3767 . . 3  |-  ( B  e.  W  ->  ( B  e.  C  <->  { B }  C_  C ) )
31, 2bi2anan9 843 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C
) ) )
4 unss 3362 . . 3  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
5 df-pr 3660 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
65sseq1i 3215 . . 3  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
74, 6bitr4i 243 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  { A ,  B }  C_  C
)
83, 7syl6bb 252 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 176    /\ wa 358    e. wcel 1696    u. cun 3163    C_ wss 3165   {csn 3653   {cpr 3654
This theorem is referenced by:  prssi  3787  lspprss  15765  lspvadd  15865  topgele  16688  prsspwg  23200  coinfliprv  23698  inttop4  25620  pgapspf2  26156  usgraedgprv  28256  usgraedgrnv  28257  dihmeetlem2N  32111
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
  Copyright terms: Public domain W3C validator