Theorem ssiun 3928

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 3149	. . . . 5 |- ( C C_ B -> ( y e. C -> y e. B ) )
2	1	reximi 2574	. . . 4 |- ( E. x e. A C C_ B -> E. x e. A ( y e. C -> y e. B ) )
3		r19.37av 2630	. . . 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 3890	. . 3 |- ( y e. U_ x e. A B <-> E. x e. A y e. B )
6	4, 5	syl6ibr 162	. 2 |- ( E. x e. A C C_ B -> ( y e. C -> y e. U_ x e. A B ) )
7	6	ssrdv 3161	1 |- ( E. x e. A C C_ B -> C C_ U_ x e. A B )

Syntax hints:   -> wi 4   e. wcel 2148   E.wrex 2456   C_ wss 3129   U_ciun 3886

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-iun 3888

This theorem is referenced by:  iunss2  3931  iunpwss  3977  iunpw  4479