ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eluni2 Unicode version

Theorem eluni2 3663
Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
eluni2  |-  ( A  e.  U. B  <->  E. x  e.  B  A  e.  x )
Distinct variable groups:    x, A    x, B

Proof of Theorem eluni2
StepHypRef Expression
1 exancom 1545 . 2  |-  ( E. x ( A  e.  x  /\  x  e.  B )  <->  E. x
( x  e.  B  /\  A  e.  x
) )
2 eluni 3662 . 2  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
3 df-rex 2366 . 2  |-  ( E. x  e.  B  A  e.  x  <->  E. x ( x  e.  B  /\  A  e.  x ) )
41, 2, 33bitr4i 211 1  |-  ( A  e.  U. B  <->  E. x  e.  B  A  e.  x )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1427    e. wcel 1439   E.wrex 2361   U.cuni 3659
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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-uni 3660
This theorem is referenced by:  uni0b  3684  intssunim  3716  iuncom4  3743  inuni  3997  ssorduni  4317  unon  4341  cnvuni  4635  chfnrn  5424  isbasis3g  11805  eltg2b  11815  tgcl  11825  epttop  11851
  Copyright terms: Public domain W3C validator