Theorem iinss2 3918

Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
iinss2	|- ( x e. A -> |^|_ x e. A B C_ B )

Proof of Theorem iinss2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2729	. . . . 5 |- y e. _V
2		eliin 3871	. . . . 5 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 5	. . . 4 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4		rsp 2513	. . . 4 |- ( A. x e. A y e. B -> ( x e. A -> y e. B ) )
5	3, 4	sylbi 120	. . 3 |- ( y e. |^|_ x e. A B -> ( x e. A -> y e. B ) )
6	5	com12 30	. 2 |- ( x e. A -> ( y e. |^|_ x e. A B -> y e. B ) )
7	6	ssrdv 3148	1 |- ( x e. A -> |^|_ x e. A B C_ B )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   |^|_ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-iin 3869

This theorem is referenced by:  dmiin  4850