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Theorem ssdif 3896
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 3746 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 598 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3 eldif 3733 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
4 eldif 3733 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
52, 3, 43imtr4g 285 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3758 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 2145  cdif 3720  wss 3723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737
This theorem is referenced by:  ssdifd  3897  php  8300  pssnn  8334  fin1a2lem13  9436  axcclem  9481  isercolllem3  14605  mvdco  18072  dprdres  18635  dpjidcl  18665  ablfac1eulem  18679  lspsnat  19359  lbsextlem2  19374  lbsextlem3  19375  mplmonmul  19679  cnsubdrglem  20012  clsconn  21454  2ndcdisj2  21481  kqdisj  21756  nulmbl2  23524  i1f1  23677  itg11  23678  itg1climres  23701  limcresi  23869  dvreslem  23893  dvres2lem  23894  dvaddbr  23921  dvmulbr  23922  lhop  23999  elqaa  24297  difres  29751  imadifxp  29752  xrge00  30026  eulerpartlemmf  30777  eulerpartlemgf  30781  bj-2upln1upl  33343  mblfinlem3  33781  mblfinlem4  33782  ismblfin  33783  cnambfre  33790  divrngidl  34159  cntzsdrg  38298  radcnvrat  39039  fourierdlem62  40902
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