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

Theorem eluni2 3612
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 1515 . 2  |-  ( E. x ( A  e.  x  /\  x  e.  B )  <->  E. x
( x  e.  B  /\  A  e.  x
) )
2 eluni 3611 . 2  |-  ( A  e.  U. B  <->  E. x
( A  e.  x  /\  x  e.  B
) )
3 df-rex 2329 . 2  |-  ( E. x  e.  B  A  e.  x  <->  E. x ( x  e.  B  /\  A  e.  x ) )
41, 2, 33bitr4i 205 1  |-  ( A  e.  U. B  <->  E. x  e.  B  A  e.  x )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   E.wrex 2324   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-uni 3609
This theorem is referenced by:  uni0b  3633  intssunim  3665  iuncom4  3692  inuni  3937  ssorduni  4241  unon  4265  cnvuni  4549  chfnrn  5306
  Copyright terms: Public domain W3C validator