Theorem elriin 3748

Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)

Assertion

Ref	Expression
elriin	|- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )

Distinct variable groups:   x, A   x, X   x, B

Allowed substitution hint:   S( x)

Proof of Theorem elriin

Step	Hyp	Ref	Expression
1		elin 3155	. 2 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ B e. |^|_ x e. X S ) )
2		eliin 3683	. . 3 |- ( B e. A -> ( B e. |^|_ x e. X S <-> A. x e. X B e. S ) )
3	2	pm5.32i 441	. 2 |- ( ( B e. A /\ B e. |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )
4	1, 3	bitri 182	1 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   A.wral 2348   i^i cin 2972   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-iin 3681

This theorem is referenced by: (None)