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Theorem unssbd 3305
Description: If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3301. Partial converse of unssd 3303. (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 3301 . . 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 3119    C_ wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134
This theorem is referenced by:  eldifpw  4462  ertr  6528  diffifi  6872  sumsplitdc  11395  fsum2dlemstep  11397  fsumabs  11428  fsumiun  11440  fprod2dlemstep  11585
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