Theorem iinss1 3928

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 3247	. . 3 |- ( A C_ B -> ( A. x e. B y e. C -> A. x e. A y e. C ) )
2		vex 2766	. . . 4 |- y e. _V
3		eliin 3921	. . . 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 3921	. . . 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 3189	1 |- ( A C_ B -> |^|_ x e. B C C_ |^|_ x e. A C )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2167   A.wral 2475   _Vcvv 2763   C_ wss 3157   |^|_ciin 3917

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-iin 3919

This theorem is referenced by: (None)