Theorem ssiun 3293

Description: Subset implication for an indexed union. (The proof was 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

Proof of Theorem ssiun

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . 5 |- (C C_ B -> (y e. C -> y e. B))
2	1	reximi 2198	. . . 4 |- (E.x e. A C C_ B -> E.x e. A (y e. C -> y e. B))
3		r19.37av 2233	. . . 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 12	. . 3 |- (E.x e. A C C_ B -> (y e. C -> E.x e. A y e. B))
5		eliun 3259	. . 3 |- (y e. U_x e. A B <-> E.x e. A y e. B)
6	4, 5	syl6ibr 230	. 2 |- (E.x e. A C C_ B -> (y e. C -> y e. U_x e. A B))
7	6	ssrdv 2622	1 |- (E.x e. A C C_ B -> C C_ U_x e. A B)

Syntax hints:   -> wi 3   e. wcel 1300  E.wrex 2106   C_ wss 2593  U_ciun 3255

This theorem is referenced by:  iunss2 3298  iunpwss 3337  iunpw 3858  onfununi 5116  oen0 5261  trcl 5752  r1tr 5765  wfrlem9 13965  neibastop2lem4 15522

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-iun 3257

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