Theorem iinss1 3690

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 3058	. . 3 |- ( A C_ B -> ( A. x e. B y e. C -> A. x e. A y e. C ) )
2		vex 2604	. . . 4 |- y e. _V
3		eliin 3683	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. B C <-> A. x e. B y e. C ) )
4	2, 3	ax-mp 7	. . 3 |- ( y e. |^|_ x e. B C <-> A. x e. B y e. C )
5		eliin 3683	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A C <-> A. x e. A y e. C ) )
6	2, 5	ax-mp 7	. . 3 |- ( y e. |^|_ x e. A C <-> A. x e. A y e. C )
7	1, 4, 6	3imtr4g 203	. 2 |- ( A C_ B -> ( y e. |^|_ x e. B C -> y e. |^|_ x e. A C ) )
8	7	ssrdv 3005	1 |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   A.wral 2348   _Vcvv 2601   C_ wss 2973   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-iin 3681

This theorem is referenced by: (None)