Theorem eluni 3813

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 2749	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2749	. . . 4 |- ( A e. x -> A e. _V )
3	2	adantr 276	. . 3 |- ( ( A e. x /\ x e. B ) -> A e. _V )
4	3	exlimiv 1598	. 2 |- ( E. x ( A e. x /\ x e. B ) -> A e. _V )
5		eleq1 2240	. . . . 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 1825	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3811	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2885	. 2 |- ( A e. _V -> ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) )
10	1, 4, 9	pm5.21nii 704	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2738   U.cuni 3810

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-uni 3811

This theorem is referenced by:  eluni2  3814  elunii  3815  eluniab  3822  uniun  3829  uniin  3830  uniss  3831  unissb  3840  dftr2  4104  unidif0  4168  unipw  4218  uniex2  4437  uniuni  4452  limom  4614  dmuni  4838  fununi  5285  nfvres  5549  elunirn  5767  tfrlem7  6318  tfrexlem  6335  tfrcldm  6364  fiuni  6977  isbasis2g  13548  tgval2  13554  ntreq0  13635  bj-uniex2  14671