Theorem elint2 3831

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 3830	. 2 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )
3		df-ral 2449	. 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 1341   e. wcel 2136   A.wral 2444   _Vcvv 2726   |^|cint 3824

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-int 3825

This theorem is referenced by:  elintg  3832  ssint  3840  intssunim  3846  iinuniss  3948  trint  4095  suplocexprlemmu  7659  suplocexprlemdisj  7661  suplocexprlemloc  7662  suplocexprlemub  7664