Theorem elint 1971

Description: Membership in class intersection.

Hypothesis

Ref	Expression
elint.1	|- A e. V

Assertion

Ref	Expression
elint	|- (A e. |^|B <-> A.x(x e. B -> A e. x))

Distinct variable group(s):   x,A   x,B

Proof of Theorem elint

Step	Hyp	Ref	Expression
1		elint.1	. 2 |- A e. V
2		eleq1 1149	. . . 4 |- (y = A -> (y e. x <-> A e. x))
3	2	imbi2d 464	. . 3 |- (y = A -> ((x e. B -> y e. x) <-> (x e. B -> A e. x)))
4	3	bialdv 935	. 2 |- (y = A -> (A.x(x e. B -> y e. x) <-> A.x(x e. B -> A e. x)))
5		df-int 1966	. 2 |- |^|B = {y | A.x(x e. B -> y e. x)}
6	1, 4, 5	elab2 1419	1 |- (A e. |^|B <-> A.x(x e. B -> A e. x))

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   e. wel 803   = wceq 1091   e. wcel 1092  Vcvv 1348  |^|cint 1965

This theorem is referenced by:  elint2 1972  elinti 1974  hbint 1975  elintab 1976  intss1 1979  intss 1983  intun 1989  intpr 1990

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-int 1966

metamath.org