Theorem elinti 3884

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 3883	. . 3 |- ( A e. |^| B -> ( A e. |^| B <-> A. x e. B A e. x ) )
2		eleq2 2260	. . . 4 |- ( x = C -> ( A e. x <-> A e. C ) )
3	2	rspccv 2865	. . 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 2167   A.wral 2475   |^|cint 3875

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-int 3876

This theorem is referenced by:  subgintm  13404  subrngintm  13844  subrgintm  13875  lssintclm  14016