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Theorem unssbd 3675
Description: If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3671. Partial converse of unssd 3673. (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 3671 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
31, 2sylibr 212 . 2  |-  ( ph  ->  ( A  C_  C  /\  B  C_  C ) )
43simprd 463 1  |-  ( ph  ->  B  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    u. cun 3467    C_ wss 3469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-v 3108  df-un 3474  df-in 3476  df-ss 3483
This theorem is referenced by:  eldifpw  6583  ertr  7316  finsschain  7816  r0weon  8379  ackbij1lem16  8604  wunfi  9088  wunex2  9105  hashf1lem2  12458  sumsplit  13532  fsum2dlem  13534  fsumabs  13564  fsumrlim  13574  fsumo1  13575  fsumiun  13584  mreexexlem3d  14890  yonedalem1  15388  yonedalem21  15389  yonedalem3a  15390  yonedalem4c  15393  yonedalem22  15394  yonedalem3b  15395  yonedainv  15397  yonffthlem  15398  ablfac1eulem  16906  lsmsp  17508  lsppratlem3  17571  mplcoe1  17891  mdetunilem9  18882  filufint  20149  fmfnfmlem4  20186  hausflim  20210  fclsfnflim  20256  fsumcn  21102  mbfeqalem  21777  itgfsum  21961  jensenlem1  23037  jensenlem2  23038  gsumvsca1  27422  gsumvsca2  27423  ordtconlem1  27528  fprod2dlem  28673  rngunsnply  30716
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