Theorem eluni2 3868

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 1632	. 2 |- ( E. x ( A e. x /\ x e. B ) <-> E. x ( x e. B /\ A e. x ) )
2		eluni 3867	. 2 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )
3		df-rex 2492	. 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 1516   e. wcel 2178   E.wrex 2487   U.cuni 3864

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-uni 3865

This theorem is referenced by:  uni0b  3889  intssunim  3921  iuncom4  3948  inuni  4215  ssorduni  4553  unon  4577  cnvuni  4882  chfnrn  5714  zrhval  14494  isbasis3g  14633  eltg2b  14641  tgcl  14651  epttop  14677  txuni2  14843