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Theorem ssun1 3234
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 701 . . 3  |-  ( x  e.  A  ->  (
x  e.  A  \/  x  e.  B )
)
2 elun 3212 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
31, 2sylibr 133 . 2  |-  ( x  e.  A  ->  x  e.  ( A  u.  B
) )
43ssriv 3096 1  |-  A  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 697    e. wcel 1480    u. cun 3064    C_ wss 3066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079
This theorem is referenced by:  ssun2  3235  ssun3  3236  elun1  3238  inabs  3303  reuun1  3353  un00  3404  undifabs  3434  undifss  3438  snsspr1  3663  snsstp1  3665  snsstp2  3666  prsstp12  3668  exmidundif  4124  sssucid  4332  unexb  4358  dmexg  4798  fvun1  5480  dftpos2  6151  tpostpos2  6155  ac6sfi  6785  caserel  6965  finomni  7005  ressxr  7802  nnssnn0  8973  un0addcl  9003  un0mulcl  9004  nn0ssxnn0  9036  fsumsplit  11169  fsum2d  11197  fsumabs  11227  ennnfonelemss  11908  bdunexb  13103
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