Theorem eluni 3811

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 2748	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2748	. . . 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 3809	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2884	. 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 2737   U.cuni 3808

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 2739  df-uni 3809

This theorem is referenced by:  eluni2  3812  elunii  3813  eluniab  3820  uniun  3827  uniin  3828  uniss  3829  unissb  3838  dftr2  4101  unidif0  4165  unipw  4215  uniex2  4434  uniuni  4449  limom  4611  dmuni  4834  fununi  5281  nfvres  5545  elunirn  5762  tfrlem7  6313  tfrexlem  6330  tfrcldm  6359  fiuni  6972  isbasis2g  13325  tgval2  13333  ntreq0  13414  bj-uniex2  14439