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Theorem ssdif 4100
Description: Difference law for subsets. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
ssdif (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem ssdif
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel 3938 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 612 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3 eldif 3921 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
4 eldif 3921 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
52, 3, 43imtr4g 296 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3951 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  cdif 3908  wss 3911
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-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928
This theorem is referenced by:  ssdifd  4101  pssnn  9115  php  9157  phpOLD  9169  pssnnOLD  9212  fin1a2lem13  10353  axcclem  10398  isercolllem3  15557  mvdco  19232  dprdres  19812  dpjidcl  19842  ablfac1eulem  19856  cntzsdrg  20283  lspsnat  20622  lbsextlem2  20636  lbsextlem3  20637  cnsubdrglem  20864  mplmonmul  21453  clsconn  22797  2ndcdisj2  22824  kqdisj  23099  nulmbl2  24916  i1f1  25070  itg11  25071  itg1climres  25095  limcresi  25265  dvreslem  25289  dvres2lem  25290  dvaddbr  25318  dvmulbr  25319  lhop  25396  elqaa  25698  difres  31564  imadifxp  31565  xrge00  31926  elrspunidl  32251  eulerpartlemmf  33032  eulerpartlemgf  33036  bj-2upln1upl  35541  pibt2  35934  mblfinlem3  36163  mblfinlem4  36164  ismblfin  36165  cnambfre  36172  divrngidl  36533  dvrelog2  40567  dvrelog3  40568  dffltz  41015  cantnftermord  41698  omabs2  41710  radcnvrat  42682  fourierdlem62  44495
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