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Theorem unssbd 3461
Description: If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3457. Partial converse of unssd 3459. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
unssad.1  |-  ( ph  ->  ( A  u.  B
)  C_  C )
Assertion
Ref Expression
unssbd  |-  ( ph  ->  B  C_  C )

Proof of Theorem unssbd
StepHypRef Expression
1 unssad.1 . . 3  |-  ( ph  ->  ( A  u.  B
)  C_  C )
2 unss 3457 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
31, 2sylibr 204 . 2  |-  ( ph  ->  ( A  C_  C  /\  B  C_  C ) )
43simprd 450 1  |-  ( ph  ->  B  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    u. cun 3254    C_ wss 3256
This theorem is referenced by:  eldifpw  4688  ertr  6849  finsschain  7341  r0weon  7820  ackbij1lem16  8041  wunfi  8522  wunex2  8539  hashf1lem2  11625  sumsplit  12472  fsum2dlem  12474  fsumabs  12500  fsumrlim  12510  fsumo1  12511  fsumiun  12520  mreexexlem3d  13791  yonedalem1  14289  yonedalem21  14290  yonedalem3a  14291  yonedalem4c  14294  yonedalem22  14295  yonedalem3b  14296  yonedainv  14298  yonffthlem  14299  ablfac1eulem  15550  lsmsp  16078  lsppratlem3  16141  mplcoe1  16448  filufint  17866  fmfnfmlem4  17903  hausflim  17927  fclsfnflim  17973  fsumcn  18764  mbfeqalem  19394  itgfsum  19578  jensenlem1  20685  jensenlem2  20686  rngunsnply  27040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-un 3261  df-in 3263  df-ss 3270
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