Theorem ssiun 3772

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 3019	. . . . 5 |- ( C C_ B -> ( y e. C -> y e. B ) )
2	1	reximi 2470	. . . 4 |- ( E. x e. A C C_ B -> E. x e. A ( y e. C -> y e. B ) )
3		r19.37av 2520	. . . 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 3734	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6	4, 5	syl6ibr 160	. 2 |- ( E. x e. A C C_ B -> ( y e. C -> y e. U_ x e. A B ) )
7	6	ssrdv 3031	1 |- ( E. x e. A C C_ B -> C C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 1438   E.wrex 2360   C_ wss 2999   U_ciun 3730

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-iun 3732

This theorem is referenced by:  iunss2  3775  iunpwss  3820  iunpw  4302