Theorem iinss2 4023

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 2805	. . . . 5 |- y e. _V
2		eliin 3975	. . . . 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 2579	. . . 4 |- ( A. x e. A y e. B -> ( x e. A -> y e. B ) )
5	3, 4	sylbi 121	. . 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 3233	1 |- ( x e. A -> |^|_ x e. A B C_ B )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   |^|_ciin 3971

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-iin 3973

This theorem is referenced by:  dmiin  4978