Theorem eluni2 3800

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 1601	. 2 |- ( E. x ( A e. x /\ x e. B ) <-> E. x ( x e. B /\ A e. x ) )
2		eluni 3799	. 2 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
3		df-rex 2454	. 2 |- ( E. x e. B A e. x <-> E. x ( x e. B /\ A e. x ) )
4	1, 2, 3	3bitr4i 211	1 |- ( A e. U. B <-> E. x e. B A e. x )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1485   e. wcel 2141   E.wrex 2449   U.cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-uni 3797

This theorem is referenced by:  uni0b  3821  intssunim  3853  iuncom4  3880  inuni  4141  ssorduni  4471  unon  4495  cnvuni  4797  chfnrn  5607  isbasis3g  12838  eltg2b  12848  tgcl  12858  epttop  12884  txuni2  13050