Theorem iinss 3953

Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iinss	|- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem iinss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2755	. . . 4 |- y e. _V
2		eliin 3906	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A B <-> A. x e. A y e. B ) )
3	1, 2	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A B <-> A. x e. A y e. B )
4		ssel 3164	. . . . 5 |- ( B C_ C -> ( y e. B -> y e. C ) )
5	4	reximi 2587	. . . 4 |- ( E. x e. A B C_ C -> E. x e. A ( y e. B -> y e. C ) )
6		r19.36av 2641	. . . 4 |- ( E. x e. A ( y e. B -> y e. C ) -> ( A. x e. A y e. B -> y e. C ) )
7	5, 6	syl 14	. . 3 |- ( E. x e. A B C_ C -> ( A. x e. A y e. B -> y e. C ) )
8	3, 7	biimtrid 152	. 2 |- ( E. x e. A B C_ C -> ( y e. |^|_ x e. A B -> y e. C ) )
9	8	ssrdv 3176	1 |- ( E. x e. A B C_ C -> |^|_ x e. A B C_ C )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2160   A.wral 2468   E.wrex 2469   _Vcvv 2752   C_ wss 3144   |^|_ciin 3902

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-iin 3904

This theorem is referenced by:  riinm  3974  reliin  4766  cnviinm  5188  iinerm  6634