Theorem elriin 4036

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 3387	. 2 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ B e. |^|_ x e. X S ) )
2		eliin 3970	. . 3 |- ( B e. A -> ( B e. |^|_ x e. X S <-> A. x e. X B e. S ) )
3	2	pm5.32i 454	. 2 |- ( ( B e. A /\ B e. |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )
4	1, 3	bitri 184	1 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2200   A.wral 2508   i^i cin 3196   |^|_ciin 3966

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-iin 3968

This theorem is referenced by: (None)