Theorem eluni 3604

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 2610	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2610	. . . 4 |- ( A e. x -> A e. _V )
3	2	adantr 270	. . 3 |- ( ( A e. x /\ x e. B ) -> A e. _V )
4	3	exlimiv 1529	. 2 |- ( E. x ( A e. x /\ x e. B ) -> A e. _V )
5		eleq1 2141	. . . . 5 |- ( y = A -> ( y e. x <-> A e. x ) )
6	5	anbi1d 452	. . . 4 |- ( y = A -> ( ( y e. x /\ x e. B ) <-> ( A e. x /\ x e. B ) ) )
7	6	exbidv 1746	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3602	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2740	. 2 |- ( A e. _V -> ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) )
10	1, 4, 9	pm5.21nii 652	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-uni 3602

This theorem is referenced by:  eluni2  3605  elunii  3606  eluniab  3613  uniun  3620  uniin  3621  uniss  3622  unissb  3631  dftr2  3877  unidif0  3941  unipw  3972  uniex2  4191  uniuni  4201  limom  4354  dmuni  4563  fununi  4987  nfvres  5227  elunirn  5426  tfrlem7  5956  tfrexlem  5971  bj-uniex2  10707