ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  unssbd GIF version

Theorem unssbd 3285
Description: If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3281. Partial converse of unssd 3283. (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 3281 . . 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 3100  wss 3102
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115
This theorem is referenced by:  eldifpw  4437  ertr  6495  diffifi  6839  sumsplitdc  11329  fsum2dlemstep  11331  fsumabs  11362  fsumiun  11374  fprod2dlemstep  11519
  Copyright terms: Public domain W3C validator