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Theorem unssbd 4189
Description: If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 4185. Partial converse of unssd 4187. (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 4185 . . 3 ((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
31, 2sylibr 233 . 2 (𝜑 → (𝐴𝐶𝐵𝐶))
43simprd 497 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  cun 3947  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966
This theorem is referenced by:  eldifpw  7755  naddcllem  8675  ertr  8718  finsschain  9359  r0weon  10007  ackbij1lem16  10230  wunfi  10716  wunex2  10733  hashf1lem2  14417  sumsplit  15714  fsum2dlem  15716  fsumabs  15747  fsumrlim  15757  fsumo1  15758  fsumiun  15767  fprod2dlem  15924  mreexexlem3d  17590  yonedalem1  18225  yonedalem21  18226  yonedalem3a  18227  yonedalem4c  18230  yonedalem22  18231  yonedalem3b  18232  yonedainv  18234  yonffthlem  18235  ablfac1eulem  19942  lsmsp  20697  lsppratlem3  20762  mplcoe1  21592  mdetunilem9  22122  filufint  23424  fmfnfmlem4  23461  hausflim  23485  fclsfnflim  23531  fsumcn  24386  itgfsum  25344  jensenlem1  26491  jensenlem2  26492  gsumvsca1  32371  gsumvsca2  32372  qsdrngilem  32608  algextdeglem1  32772  ordtconnlem1  32904  vhmcls  34557  mclsppslem  34574  rngunsnply  41915  brtrclfv2  42478
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