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Theorem ssdif 3729
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 3582 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 587 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3 eldif 3570 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
4 eldif 3570 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
52, 3, 43imtr4g 285 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3594 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1987  cdif 3557  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574
This theorem is referenced by:  ssdifd  3730  php  8104  pssnn  8138  fin1a2lem13  9194  axcclem  9239  isercolllem3  14347  mvdco  17805  dprdres  18367  dpjidcl  18397  ablfac1eulem  18411  lspsnat  19085  lbsextlem2  19099  lbsextlem3  19100  mplmonmul  19404  cnsubdrglem  19737  clsconn  21173  2ndcdisj2  21200  kqdisj  21475  nulmbl2  23244  i1f1  23397  itg11  23398  itg1climres  23421  limcresi  23589  dvreslem  23613  dvres2lem  23614  dvaddbr  23641  dvmulbr  23642  lhop  23717  elqaa  24015  difres  29299  imadifxp  29300  xrge00  29513  eulerpartlemmf  30260  eulerpartlemgf  30264  bj-2upln1upl  32712  mblfinlem3  33119  mblfinlem4  33120  ismblfin  33121  cnambfre  33129  divrngidl  33498  cntzsdrg  37292  radcnvrat  38034  fourierdlem62  39722
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