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Theorem unssad 3178
Description: If  ( A  u.  B ) is contained in  C, so is  A. One-way deduction form of unss 3175. Partial converse of unssd 3177. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
unssad.1  |-  ( ph  ->  ( A  u.  B
)  C_  C )
Assertion
Ref Expression
unssad  |-  ( ph  ->  A  C_  C )

Proof of Theorem unssad
StepHypRef Expression
1 unssad.1 . . 3  |-  ( ph  ->  ( A  u.  B
)  C_  C )
2 unss 3175 . . 3  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
31, 2sylibr 133 . 2  |-  ( ph  ->  ( A  C_  C  /\  B  C_  C ) )
43simpld 111 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    u. cun 2998    C_ wss 3000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013
This theorem is referenced by:  ersym  6318  findcard2d  6661  findcard2sd  6662  diffifi  6664  sumsplitdc  10887  fsumabs  10920  fsumiun  10932
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