Theorem ssiun 3863

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem ssiun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3096	. . . . 5 |- ( C C_ B -> ( y e. C -> y e. B ) )
2	1	reximi 2532	. . . 4 |- ( E. x e. A C C_ B -> E. x e. A ( y e. C -> y e. B ) )
3		r19.37av 2587	. . . 4 |- ( E. x e. A ( y e. C -> y e. B ) -> ( y e. C -> E. x e. A y e. B ) )
4	2, 3	syl 14	. . 3 |- ( E. x e. A C C_ B -> ( y e. C -> E. x e. A y e. B ) )
5		eliun 3825	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6	4, 5	syl6ibr 161	. 2 |- ( E. x e. A C C_ B -> ( y e. C -> y e. U_ x e. A B ) )
7	6	ssrdv 3108	1 |- ( E. x e. A C C_ B -> C C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 1481   E.wrex 2418   C_ wss 3076   U_ciun 3821

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-iun 3823

This theorem is referenced by:  iunss2  3866  iunpwss  3912  iunpw  4409