Theorem eluni2 3605

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

Step	Hyp	Ref	Expression
1		exancom 1539	. 2 |- ( E. x ( A e. x /\ x e. B ) <-> E. x ( x e. B /\ A e. x ) )
2		eluni 3604	. 2 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
3		df-rex 2354	. 2 |- ( E. x e. B A e. x <-> E. x ( x e. B /\ A e. x ) )
4	1, 2, 3	3bitr4i 210	1 |- ( A e. U. B <-> E. x e. B A e. x )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   E.wrex 2349   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-uni 3602

This theorem is referenced by:  uni0b  3626  intssunim  3658  iuncom4  3685  inuni  3930  ssorduni  4231  unon  4255  cnvuni  4539  chfnrn  5299