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Theorem unssbd 3178
Description: If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3174. Partial converse of unssd 3176. (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 3174 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
31, 2sylibr 132 . 2  |-  ( ph  ->  ( A  C_  C  /\  B  C_  C ) )
43simprd 112 1  |-  ( ph  ->  B  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    u. cun 2997    C_ wss 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012
This theorem is referenced by:  eldifpw  4299  ertr  6307  diffifi  6610  sumsplitdc  10826  fsum2dlemstep  10828  fsumabs  10859  fsumiun  10871
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