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Theorem ssdif 3728
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 3581 . . . 4 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21anim1d 587 . . 3 (𝐴𝐵 → ((𝑥𝐴 ∧ ¬ 𝑥𝐶) → (𝑥𝐵 ∧ ¬ 𝑥𝐶)))
3 eldif 3569 . . 3 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐶))
4 eldif 3569 . . 3 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵 ∧ ¬ 𝑥𝐶))
52, 3, 43imtr4g 285 . 2 (𝐴𝐵 → (𝑥 ∈ (𝐴𝐶) → 𝑥 ∈ (𝐵𝐶)))
65ssrdv 3593 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 1987  cdif 3556  wss 3559
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 3191  df-dif 3562  df-in 3566  df-ss 3573
This theorem is referenced by:  ssdifd  3729  php  8095  pssnn  8129  fin1a2lem13  9185  axcclem  9230  isercolllem3  14338  mvdco  17793  dprdres  18355  dpjidcl  18385  ablfac1eulem  18399  lspsnat  19073  lbsextlem2  19087  lbsextlem3  19088  mplmonmul  19392  cnsubdrglem  19725  clsconn  21152  2ndcdisj2  21179  kqdisj  21454  nulmbl2  23223  i1f1  23376  itg11  23377  itg1climres  23400  limcresi  23568  dvreslem  23592  dvres2lem  23593  dvaddbr  23620  dvmulbr  23621  lhop  23696  elqaa  23994  difres  29276  imadifxp  29277  xrge00  29489  eulerpartlemmf  30236  eulerpartlemgf  30240  bj-2upln1upl  32686  mblfinlem3  33107  mblfinlem4  33108  ismblfin  33109  cnambfre  33117  divrngidl  33486  cntzsdrg  37280  radcnvrat  38022  fourierdlem62  39713
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