Theorem elriin 3891

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 3264	. 2 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ B e. |^|_ x e. X S ) )
2		eliin 3826	. . 3 |- ( B e. A -> ( B e. |^|_ x e. X S <-> A. x e. X B e. S ) )
3	2	pm5.32i 450	. 2 |- ( ( B e. A /\ B e. |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )
4	1, 3	bitri 183	1 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 1481   A.wral 2417   i^i cin 3075   |^|_ciin 3822

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3082  df-iin 3824

This theorem is referenced by: (None)