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Theorem ssun1 3298
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 3276 . . 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 3159 1  |-  A  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 708    e. wcel 2148    u. cun 3127    C_ wss 3129
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 2739  df-un 3133  df-in 3135  df-ss 3142
This theorem is referenced by:  ssun2  3299  ssun3  3300  elun1  3302  inabs  3367  reuun1  3417  un00  3469  undifabs  3499  undifss  3503  snsspr1  3740  snsstp1  3742  snsstp2  3743  prsstp12  3745  exmidundif  4206  sssucid  4415  unexb  4442  dmexg  4891  fvun1  5582  dftpos2  6261  tpostpos2  6265  ac6sfi  6897  caserel  7085  finomni  7137  ressxr  8000  nnssnn0  9178  un0addcl  9208  un0mulcl  9209  nn0ssxnn0  9241  fsumsplit  11414  fsum2d  11442  fsumabs  11472  fprodsplitdc  11603  fprod2d  11630  ennnfonelemss  12410  cnfldbas  13429  cnfldadd  13430  cnfldmul  13431  lgsdir2lem3  14401  bdunexb  14642
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