Theorem iunss1 3884

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 3212	. . 3 |- ( A C_ B -> ( E. x e. A y e. C -> E. x e. B y e. C ) )
2		eliun 3877	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
3		eliun 3877	. . 3 |- ( y e. U_ x e. B C <-> E. x e. B y e. C )
4	1, 2, 3	3imtr4g 204	. 2 |- ( A C_ B -> ( y e. U_ x e. A C -> y e. U_ x e. B C ) )
5	4	ssrdv 3153	1 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Syntax hints:   -> wi 4   e. wcel 2141   E.wrex 2449   C_ wss 3121   U_ciun 3873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iun 3875

This theorem is referenced by:  iuneq1  3886  iunxdif2  3921  fsumiun  11440