Theorem iunss1 3981

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 3292	. . 3 |- ( A C_ B -> ( E. x e. A y e. C -> E. x e. B y e. C ) )
2		eliun 3974	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
3		eliun 3974	. . 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 3233	1 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Syntax hints:   -> wi 4   e. wcel 2202   E.wrex 2511   C_ wss 3200   U_ciun 3970

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-iun 3972

This theorem is referenced by:  iuneq1  3983  iunxdif2  4019  fsumiun  12037