Theorem iinss2 3980

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 2775	. . . . 5 |- y e. _V
2		eliin 3932	. . . . 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 2553	. . . 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 3199	1 |- ( x e. A -> |^|_ x e. A B C_ B )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   |^|_ciin 3928

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-iin 3930

This theorem is referenced by:  dmiin  4924