Theorem eluni2 3854

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 1631	. 2 |- ( E. x ( A e. x /\ x e. B ) <-> E. x ( x e. B /\ A e. x ) )
2		eluni 3853	. 2 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
3		df-rex 2490	. 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 1515   e. wcel 2176   E.wrex 2485   U.cuni 3850

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-uni 3851

This theorem is referenced by:  uni0b  3875  intssunim  3907  iuncom4  3934  inuni  4199  ssorduni  4535  unon  4559  cnvuni  4864  chfnrn  5691  zrhval  14379  isbasis3g  14518  eltg2b  14526  tgcl  14536  epttop  14562  txuni2  14728