Theorem elriin 4041

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 3390	. 2 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ B e. |^|_ x e. X S ) )
2		eliin 3975	. . 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 2202   A.wral 2510   i^i cin 3199   |^|_ciin 3971

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-iin 3973

This theorem is referenced by: (None)