Theorem elint2 3838

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Hypothesis

Ref	Expression
elint2.1	|- A e. _V

Assertion

Ref	Expression
elint2	|- ( A e. |^| B <-> A. x e. B A e. x )

Distinct variable groups:   x, A   x, B

Proof of Theorem elint2

Step	Hyp	Ref	Expression
1		elint2.1	. . 3 |- A e. _V
2	1	elint 3837	. 2 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )
3		df-ral 2453	. 2 |- ( A. x e. B A e. x <-> A. x ( x e. B -> A e. x ) )
4	2, 3	bitr4i 186	1 |- ( A e. |^| B <-> A. x e. B A e. x )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   e. wcel 2141   A.wral 2448   _Vcvv 2730   |^|cint 3831

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-int 3832

This theorem is referenced by:  elintg  3839  ssint  3847  intssunim  3853  iinuniss  3955  trint  4102  suplocexprlemmu  7680  suplocexprlemdisj  7682  suplocexprlemloc  7683  suplocexprlemub  7685