Theorem iinss1 3977

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 3288	. . 3 |- ( A C_ B -> ( A. x e. B y e. C -> A. x e. A y e. C ) )
2		vex 2802	. . . 4 |- y e. _V
3		eliin 3970	. . . 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 3970	. . . 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 205	. 2 |- ( A C_ B -> ( y e. |^|_ x e. B C -> y e. |^|_ x e. A C ) )
8	7	ssrdv 3230	1 |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   |^|_ciin 3966

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-iin 3968

This theorem is referenced by: (None)