Theorem elinti 3935

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 3934	. . 3 |- (A e. |^|B -> (A e. |^|B <-> A.x e. B A e. x))
2		eleq2 2414	. . . 4 |- (x = C -> (A e. x <-> A e. C))
3	2	rspccv 2952	. . 3 |- (A.x e. B A e. x -> (C e. B -> A e. C))
4	1, 3	syl6bi 219	. 2 |- (A e. |^|B -> (A e. |^|B -> (C e. B -> A e. C)))
5	4	pm2.43i 43	1 |- (A e. |^|B -> (C e. B -> A e. C))

Syntax hints:   -> wi 4   e. wcel 1710  A.wral 2614  |^|cint 3926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-int 3927

This theorem is referenced by: (None)