Theorem ssiun2 3295

Description: Identity law for subset of an indexed union. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ssiun2	|- (x e. A -> B C_ U_x e. A B)

Proof of Theorem ssiun2

Step	Hyp	Ref	Expression
1		ra4e 2156	. . . 4 |- ((x e. A /\ y e. B) -> E.x e. A y e. B)
2	1	ex 402	. . 3 |- (x e. A -> (y e. B -> E.x e. A y e. B))
3		eliun 3259	. . 3 |- (y e. U_x e. A B <-> E.x e. A y e. B)
4	2, 3	syl6ibr 230	. 2 |- (x e. A -> (y e. B -> y e. U_x e. A B))
5	4	ssrdv 2622	1 |- (x e. A -> B 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:  ssiun2s 3297  ixpf 5415  r1val1 5769  rankuni2 5801  rankval4 5813  cplem1 5850  omsubsdomlem2 5880  elomsubsd 5885  infxpidmlem5 8825  bnj894 13327  bnj999 13365  bnj1014 13374  bnj1137 13433  bnj1404 13517  omsubsdomlem2OLD 15389  elomsubsdOLD 15394

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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