Theorem eluni 3938

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 2833	. 2 |- ( A e. U. B -> A e. _V )
2		elex 2833	. . . 4 |- ( A e. x -> A e. _V )
3	2	adantr 276	. . 3 |- ( ( A e. x /\ x e. B ) -> A e. _V )
4	3	exlimiv 1651	. 2 |- ( E. x ( A e. x /\ x e. B ) -> A e. _V )
5		eleq1 2301	. . . . 5 |- ( y = A -> ( y e. x <-> A e. x ) )
6	5	anbi1d 469	. . . 4 |- ( y = A -> ( ( y e. x /\ x e. B ) <-> ( A e. x /\ x e. B ) ) )
7	6	exbidv 1878	. . 3 |- ( y = A -> ( E. x ( y e. x /\ x e. B ) <-> E. x ( A e. x /\ x e. B ) ) )
8		df-uni 3936	. . 3 |- U. B = { y | E. x ( y e. x /\ x e. B ) }
9	7, 8	elab2g 2973	. 2 |- ( A e. _V -> ( A e. U. B <-> E. x ( A e. x /\ x e. B ) ) )
10	1, 4, 9	pm5.21nii 716	1 |- ( A e. U. B <-> E. x ( A e. x /\ x e. B ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-uni 3936

This theorem is used by:  eluni2  3939  elunii  3940  eluniab  3947  uniun  3954  uniin  3955  uniss  3956  unissb  3965  dftr2  4231  unidif0  4304  unipw  4357  uniex2  4581  uniex2OLD  4582  uniuni  4597  limom  4761  dmuni  4991  fununi  5449  nfvres  5732  elunirn  5972  tfrlem7  6588  tfrexlem  6605  tfrcldm  6634  fiuni  7312  isbasis2g  15146  tgval2  15152  ntreq0  15233  bj-uniex2  16942