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Theorem unssbd 3305
Description: If (𝐴𝐵) 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 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
Assertion
Ref Expression
unssbd (𝜑𝐵𝐶)

Proof of Theorem unssbd
StepHypRef Expression
1 unssad.1 . . 3 (𝜑 → (𝐴𝐵) ⊆ 𝐶)
2 unss 3301 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
31, 2sylibr 133 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
43simprd 113 1 (𝜑𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  cun 3119  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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