Theorem elint2 3849

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 3848	. 2 |- ( A e. |^| B <-> A. x ( x e. B -> A e. x ) )
3		df-ral 2460	. 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 1351   e. wcel 2148   A.wral 2455   _Vcvv 2737   |^|cint 3842

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-int 3843

This theorem is referenced by:  elintg  3850  ssint  3858  intssunim  3864  iinuniss  3966  trint  4113  suplocexprlemmu  7695  suplocexprlemdisj  7697  suplocexprlemloc  7698  suplocexprlemub  7700