Theorem iinss2 3934

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 2738	. . . . 5 |- y e. _V
2		eliin 3887	. . . . 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 2522	. . . 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 3159	1 |- ( x e. A -> |^|_ x e. A B C_ B )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2146   A.wral 2453   _Vcvv 2735   C_ wss 3127   |^|_ciin 3883

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-iin 3885

This theorem is referenced by:  dmiin  4866