Theorem elinti 3894

Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elinti	|- ( A e. |^| B -> ( C e. B -> A e. C ) )

Proof of Theorem elinti

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintg 3893	. . 3 |- ( A e. |^| B -> ( A e. |^| B <-> A. x e. B A e. x ) )
2		eleq2 2269	. . . 4 |- ( x = C -> ( A e. x <-> A e. C ) )
3	2	rspccv 2874	. . 3 |- ( A. x e. B A e. x -> ( C e. B -> A e. C ) )
4	1, 3	biimtrdi 163	. 2 |- ( A e. |^| B -> ( A e. |^| B -> ( C e. B -> A e. C ) ) )
5	4	pm2.43i 49	1 |- ( A e. |^| B -> ( C e. B -> A e. C ) )

Syntax hints:   -> wi 4   e. wcel 2176   A.wral 2484   |^|cint 3885

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-int 3886

This theorem is referenced by:  subgintm  13534  subrngintm  13974  subrgintm  14005  lssintclm  14146