Theorem elrint 4005

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 3412	. 2 |- ( X e. ( A i^i |^| B ) <-> ( X e. A /\ X e. |^| B ) )
2		elintg 3973	. . 3 |- ( X e. A -> ( X e. |^| B <-> A. y e. B X e. y ) )
3	2	pm5.32i 458	. 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 2209   A.wral 2528   i^i cin 3219   |^|cint 3965

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-in 3226  df-int 3966

This theorem is referenced by:  elrint2  4006