Theorem elrint 3910

Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
elrint	|- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )

Distinct variable groups:   y, B   y, X

Allowed substitution hint:   A( y)

Proof of Theorem elrint

Step	Hyp	Ref	Expression
1		elin 3342	. 2 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ X e. |^| B ) )
2		elintg 3878	. . 3 |- ( X e. A -> ( X e. |^| B <-> A. y e. B X e. y ) )
3	2	pm5.32i 454	. 2 |- ( ( X e. A /\ X e. |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )
4	1, 3	bitri 184	1 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ A. y e. B X e. y ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2164   A.wral 2472   i^i cin 3152   |^|cint 3870

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-int 3871

This theorem is referenced by:  elrint2  3911