ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prssg Unicode version

Theorem prssg 3643
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 3622 . . 3  |-  ( A  e.  V  ->  ( A  e.  C  <->  { A }  C_  C ) )
2 snssg 3622 . . 3  |-  ( B  e.  W  ->  ( B  e.  C  <->  { B }  C_  C ) )
31, 2bi2anan9 578 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C
) ) )
4 unss 3216 . . 3  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
5 df-pr 3500 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
65sseq1i 3089 . . 3  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
74, 6bitr4i 186 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  { A ,  B }  C_  C
)
83, 7syl6bb 195 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    /\ wa 103    <-> wb 104    e. wcel 1463    u. cun 3035    C_ wss 3037   {csn 3493   {cpr 3494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500
This theorem is referenced by:  prssi  3644  prsspwg  3645  topgele  12036
  Copyright terms: Public domain W3C validator