Theorem ssiun 3915

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 3141	. . . . 5 |- ( C C_ B -> ( y e. C -> y e. B ) )
2	1	reximi 2567	. . . 4 |- ( E. x e. A C C_ B -> E. x e. A ( y e. C -> y e. B ) )
3		r19.37av 2623	. . . 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 3877	. . 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 3153	1 |- ( E. x e. A C C_ B -> C C_ U_ x e. A B )

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:  iunss2  3918  iunpwss  3964  iunpw  4465