Theorem iinss1 3820

Description: Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Assertion

Ref	Expression
iinss1	|- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iinss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 3156	. . 3 |- ( A C_ B -> ( A. x e. B y e. C -> A. x e. A y e. C ) )
2		vex 2684	. . . 4 |- y e. _V
3		eliin 3813	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. B C <-> A. x e. B y e. C ) )
4	2, 3	ax-mp 5	. . 3 |- ( y e. |^|_ x e. B C <-> A. x e. B y e. C )
5		eliin 3813	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A C <-> A. x e. A y e. C ) )
6	2, 5	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A C <-> A. x e. A y e. C )
7	1, 4, 6	3imtr4g 204	. 2 |- ( A C_ B -> ( y e. |^|_ x e. B C -> y e. |^|_ x e. A C ) )
8	7	ssrdv 3098	1 |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 1480   A.wral 2414   _Vcvv 2681   C_ wss 3066   |^|_ciin 3809

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-iin 3811

This theorem is referenced by: (None)