Theorem iunss1 3689

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 3059	. . 3 |- ( A C_ B -> ( E. x e. A y e. C -> E. x e. B y e. C ) )
2		eliun 3682	. . 3 |- ( y e. U_ x e. A C <-> E. x e. A y e. C )
3		eliun 3682	. . 3 |- ( y e. U_ x e. B C <-> E. x e. B y e. C )
4	1, 2, 3	3imtr4g 203	. 2 |- ( A C_ B -> ( y e. U_ x e. A C -> y e. U_ x e. B C ) )
5	4	ssrdv 3005	1 |- ( A C_ B -> U_ x e. A C C_ U_ x e. B C )

Syntax hints:   -> wi 4   e. wcel 1433   E.wrex 2349   C_ wss 2973   U_ciun 3678

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-iun 3680

This theorem is referenced by:  iuneq1  3691  iunxdif2  3726