Theorem iunss1 3976

Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunss1	|- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem iunss1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 3289	. . 3 |- ( A C_ B -> ( E. x e. A y e. C -> E. x e. B y e. C ) )
2		eliun 3969	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
3		eliun 3969	. . 3 |- ( y e. U_ x e. B C <-> E. x e. B y e. C )
4	1, 2, 3	3imtr4g 205	. 2 |- ( A C_ B -> ( y e. U_ x e. A C -> y e. U_ x e. B C ) )
5	4	ssrdv 3230	1 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Syntax hints:   -> wi 4   e. wcel 2200   E.wrex 2509   C_ wss 3197   U_ciun 3965

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-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-iun 3967

This theorem is referenced by:  iuneq1  3978  iunxdif2  4014  fsumiun  11988