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Theorem grothprimlem 8455
Description: Lemma for grothprim 8456. 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 3656 . . 3  |-  { u ,  v }  =  { h  |  (
h  =  u  \/  h  =  v ) }
21eleq1i 2346 . 2  |-  ( { u ,  v }  e.  w  <->  { h  |  ( h  =  u  \/  h  =  v ) }  e.  w )
3 clabel 2404 . 2  |-  ( { h  |  ( h  =  u  \/  h  =  v ) }  e.  w  <->  E. g
( g  e.  w  /\  A. h ( h  e.  g  <->  ( h  =  u  \/  h  =  v ) ) ) )
42, 3bitri 240 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 176    \/ wo 357    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   {cpr 3641
This theorem is referenced by:  grothprim  8456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647
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