Theorem eluni 3775

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 2723	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2723	. . . 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 2220	. . . . 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 1805	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3773	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2859	. 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 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712   U.cuni 3772

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-uni 3773

This theorem is referenced by:  eluni2  3776  elunii  3777  eluniab  3784  uniun  3791  uniin  3792  uniss  3793  unissb  3802  dftr2  4064  unidif0  4128  unipw  4177  uniex2  4396  uniuni  4411  limom  4573  dmuni  4796  fununi  5238  nfvres  5501  elunirn  5716  tfrlem7  6264  tfrexlem  6281  tfrcldm  6310  fiuni  6922  isbasis2g  12454  tgval2  12462  ntreq0  12543  bj-uniex2  13502