Theorem eluni 3852

Description: Membership in class union. (Contributed by NM, 22-May-1994.)

Assertion

Ref	Expression
eluni	|- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem eluni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2782	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2782	. . . 4 |- ( A e. x -> A e. _V )
3	2	adantr 276	. . 3 |- ( ( A e. x /\ x e. B ) -> A e. _V )
4	3	exlimiv 1620	. 2 |- ( E. x ( A e. x /\ x e. B ) -> A e. _V )
5		eleq1 2267	. . . . 5 |- ( y = A -> ( y e. x <-> A e. x ) )
6	5	anbi1d 465	. . . 4 |- ( y = A -> ( ( y e. x /\ x e. B ) <-> ( A e. x /\ x e. B ) ) )
7	6	exbidv 1847	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3850	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2919	. 2 |- ( A e. _V -> ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) )
10	1, 4, 9	pm5.21nii 705	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1372   E.wex 1514   e. wcel 2175   _Vcvv 2771   U.cuni 3849

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-uni 3850

This theorem is referenced by:  eluni2  3853  elunii  3854  eluniab  3861  uniun  3868  uniin  3869  uniss  3870  unissb  3879  dftr2  4143  unidif0  4210  unipw  4260  uniex2  4482  uniuni  4497  limom  4661  dmuni  4887  fununi  5341  nfvres  5609  elunirn  5834  tfrlem7  6402  tfrexlem  6419  tfrcldm  6448  fiuni  7079  isbasis2g  14488  tgval2  14494  ntreq0  14575  bj-uniex2  15814