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Theorem prssg 4170
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 4148 . . 3  |-  ( A  e.  V  ->  ( A  e.  C  <->  { A }  C_  C ) )
2 snssg 4148 . . 3  |-  ( B  e.  W  ->  ( B  e.  C  <->  { B }  C_  C ) )
31, 2bi2anan9 873 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  ( { A }  C_  C  /\  { B }  C_  C
) ) )
4 unss 3663 . . 3  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  ( { A }  u.  { B } )  C_  C
)
5 df-pr 4017 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
65sseq1i 3513 . . 3  |-  ( { A ,  B }  C_  C  <->  ( { A }  u.  { B } )  C_  C
)
74, 6bitr4i 252 . 2  |-  ( ( { A }  C_  C  /\  { B }  C_  C )  <->  { A ,  B }  C_  C
)
83, 7syl6bb 261 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e.  C  /\  B  e.  C )  <->  { A ,  B }  C_  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1804    u. cun 3459    C_ wss 3461   {csn 4014   {cpr 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-un 3466  df-in 3468  df-ss 3475  df-sn 4015  df-pr 4017
This theorem is referenced by:  prssi  4171  prsspwg  4172  lspprss  17617  lspvadd  17721  topgele  19413  usgraedgprv  24354  usgraedgrnv  24355  usgraedg4  24365  2trllemH  24532  2trllemE  24533  fourierdlem20  31863  fourierdlem50  31893  fourierdlem54  31897  fourierdlem64  31907  fourierdlem76  31919  prelpw  32253  dihmeetlem2N  36901
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