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Theorem ssun1 3299
Description: Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
ssun1  |-  A  C_  ( A  u.  B
)

Proof of Theorem ssun1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 orc 712 . . 3  |-  ( x  e.  A  ->  (
x  e.  A  \/  x  e.  B )
)
2 elun 3277 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
31, 2sylibr 134 . 2  |-  ( x  e.  A  ->  x  e.  ( A  u.  B
) )
43ssriv 3160 1  |-  A  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 708    e. wcel 2148    u. cun 3128    C_ wss 3130
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-v 2740  df-un 3134  df-in 3136  df-ss 3143
This theorem is referenced by:  ssun2  3300  ssun3  3301  elun1  3303  inabs  3368  reuun1  3418  un00  3470  undifabs  3500  undifss  3504  snsspr1  3741  snsstp1  3743  snsstp2  3744  prsstp12  3746  exmidundif  4207  sssucid  4416  unexb  4443  dmexg  4892  fvun1  5583  dftpos2  6262  tpostpos2  6266  ac6sfi  6898  caserel  7086  finomni  7138  ressxr  8001  nnssnn0  9179  un0addcl  9209  un0mulcl  9210  nn0ssxnn0  9242  fsumsplit  11415  fsum2d  11443  fsumabs  11473  fprodsplitdc  11604  fprod2d  11631  ennnfonelemss  12411  cnfldbas  13462  cnfldadd  13463  cnfldmul  13464  lgsdir2lem3  14434  bdunexb  14675
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