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Theorem grothprimlem 8708
Description: Lemma for grothprim 8709. Expand the membership of an unordered pair into primitives. (Contributed by NM, 29-Mar-2007.)
Assertion
Ref Expression
grothprimlem  |-  ( { u ,  v }  e.  w  <->  E. g
( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v ) ) ) )
Distinct variable group:    w, v, u, h, g

Proof of Theorem grothprimlem
StepHypRef Expression
1 dfpr2 3830 . . 3  |-  { u ,  v }  =  { h  |  (
h  =  u  \/  h  =  v ) }
21eleq1i 2499 . 2  |-  ( { u ,  v }  e.  w  <->  { h  |  ( h  =  u  \/  h  =  v ) }  e.  w )
3 clabel 2557 . 2  |-  ( { h  |  ( h  =  u  \/  h  =  v ) }  e.  w  <->  E. g
( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v ) ) ) )
42, 3bitri 241 1  |-  ( { u ,  v }  e.  w  <->  E. g
( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359   A.wal 1549   E.wex 1550    e. wcel 1725   {cab 2422   {cpr 3815
This theorem is referenced by:  grothprim  8709
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821
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