Theorem elinti 3977

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 3976	. . 3 |- ( A e. |^| B -> ( A e. |^| B <-> A. x e. B A e. x ) )
2		eleq2 2302	. . . 4 |- ( x = C -> ( A e. x <-> A e. C ) )
3	2	rspccv 2926	. . 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 2209   A.wral 2528   |^|cint 3968

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-int 3969

This theorem is referenced by:  subgintm  13981  subrngintm  14496  subrgintm  14527  lssintclm  14696