Theorem elriin 3727

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 3126	. 2 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ B e. |^|_ x e. X S ) )
2		eliin 3662	. . 3 |- ( B e. A -> ( B e. |^|_ x e. X S <-> A. x e. X B e. S ) )
3	2	pm5.32i 427	. 2 |- ( ( B e. A /\ B e. |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )
4	1, 3	bitri 173	1 |- ( B e. ( A i^i |^|_ x e. X S ) <-> ( B e. A /\ A. x e. X B e. S ) )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   A.wral 2306   i^i cin 2916   |^|_ciin 3658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-iin 3660

This theorem is referenced by: (None)