Theorem elint2 3936

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 3935	. 2 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )
3		df-ral 2514	. 2 |- ( A. x e. B A e. x <-> A. x ( x e. B -> A e. x ) )
4	2, 3	bitr4i 187	1 |- ( A e. |^| B <-> A. x e. B A e. x )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1395   e. wcel 2201   A.wral 2509   _Vcvv 2801   |^|cint 3929

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-v 2803  df-int 3930

This theorem is referenced by:  elintg  3937  ssint  3945  intssunim  3951  iinuniss  4054  trint  4203  suplocexprlemmu  7943  suplocexprlemdisj  7945  suplocexprlemloc  7946  suplocexprlemub  7948