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Theorem unssbd 3249
Description: If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3245. Partial converse of unssd 3247. (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 3245 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
31, 2sylibr 133 . 2  |-  ( ph  ->  ( A  C_  C  /\  B  C_  C ) )
43simprd 113 1  |-  ( ph  ->  B  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    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:  eldifpw  4393  ertr  6437  diffifi  6781  sumsplitdc  11194  fsum2dlemstep  11196  fsumabs  11227  fsumiun  11239
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