Theorem eluni 3747

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 2700	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2700	. . . 4 |- ( A e. x -> A e. _V )
3	2	adantr 274	. . 3 |- ( ( A e. x /\ x e. B ) -> A e. _V )
4	3	exlimiv 1578	. 2 |- ( E. x ( A e. x /\ x e. B ) -> A e. _V )
5		eleq1 2203	. . . . 5 |- ( y = A -> ( y e. x <-> A e. x ) )
6	5	anbi1d 461	. . . 4 |- ( y = A -> ( ( y e. x /\ x e. B ) <-> ( A e. x /\ x e. B ) ) )
7	6	exbidv 1798	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3745	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2835	. 2 |- ( A e. _V -> ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) )
10	1, 4, 9	pm5.21nii 694	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689   U.cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-uni 3745

This theorem is referenced by:  eluni2  3748  elunii  3749  eluniab  3756  uniun  3763  uniin  3764  uniss  3765  unissb  3774  dftr2  4036  unidif0  4099  unipw  4147  uniex2  4366  uniuni  4380  limom  4535  dmuni  4757  fununi  5199  nfvres  5462  elunirn  5675  tfrlem7  6222  tfrexlem  6239  tfrcldm  6268  fiuni  6874  isbasis2g  12251  tgval2  12259  ntreq0  12340  bj-uniex2  13285