Theorem iunss1 4018

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 3313	. . 3 |- ( A C_ B -> ( E. x e. A y e. C -> E. x e. B y e. C ) )
2		eliun 4011	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
3		eliun 4011	. . 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 3254	1 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Syntax hints:   -> wi 4   e. wcel 2209   E.wrex 2529   C_ wss 3220   U_ciun 4007

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-iun 4009

This theorem is referenced by:  iuneq1  4020  iunxdif2  4056  fsumiun  12222