Theorem eluni2 3918

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 1657	. 2 |- ( E. x ( A e. x /\ x e. B ) <-> E. x ( x e. B /\ A e. x ) )
2		eluni 3917	. 2 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
3		df-rex 2526	. 2 |- ( E. x e. B A e. x <-> E. x ( x e. B /\ A e. x ) )
4	1, 2, 3	3bitr4i 212	1 |- ( A e. U. B <-> E. x e. B A e. x )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1541   e. wcel 2203   E.wrex 2521   U.cuni 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-uni 3915

This theorem is referenced by:  uni0b  3939  intssunim  3971  iuncom4  3998  inuni  4267  ssorduni  4609  unon  4633  cnvuni  4941  chfnrn  5789  zrhval  14765  isbasis3g  14911  eltg2b  14919  tgcl  14929  epttop  14955  txuni2  15121