Theorem eluni 3896

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 2814	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2814	. . . 4 |- ( A e. x -> A e. _V )
3	2	adantr 276	. . 3 |- ( ( A e. x /\ x e. B ) -> A e. _V )
4	3	exlimiv 1646	. 2 |- ( E. x ( A e. x /\ x e. B ) -> A e. _V )
5		eleq1 2294	. . . . 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 1873	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3894	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2953	. 2 |- ( A e. _V -> ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) )
10	1, 4, 9	pm5.21nii 711	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   U.cuni 3893

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-uni 3894

This theorem is referenced by:  eluni2  3897  elunii  3898  eluniab  3905  uniun  3912  uniin  3913  uniss  3914  unissb  3923  dftr2  4189  unidif0  4257  unipw  4309  uniex2  4533  uniuni  4548  limom  4712  dmuni  4941  fununi  5398  nfvres  5675  elunirn  5906  tfrlem7  6482  tfrexlem  6499  tfrcldm  6528  fiuni  7176  isbasis2g  14768  tgval2  14774  ntreq0  14855  bj-uniex2  16511