Theorem ssdifd 3708

Description: If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 3707. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
ssdifd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssdifd	⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Proof of Theorem ssdifd

Step	Hyp	Ref	Expression
1		ssdifd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ssdif 3707	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Syntax hints:   → wi 4   ∖ cdif 3537   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554

This theorem is referenced by:  ssdif2d  3711  domunsncan  7945  fin1a2lem13  9117  seqcoll2  13106  rpnnen2lem11  14792  coprmprod  15213  mrieqv2d  16122  mrissmrid  16124  mreexexlem4d  16130  acsfiindd  17000  lsppratlem3  18970  lsppratlem4  18971  f1lindf  19980  lpss3  20758  lpcls  20978  fin1aufil  21546  rrxmval  22996  rrxmetlem  22998  uniioombllem3  23159  i1fmul  23269  itg1addlem4  23272  itg1climres  23287  limciun  23464  ig1peu  23735  ig1pdvds  23740  nbgrassovt  25964  usgreghash2spotv  26593  sitgclg  29731  mthmpps  30733  poimirlem11  32590  poimirlem12  32591  poimirlem15  32594  dochfln0  35784  lcfl6  35807  lcfrlem16  35865  hdmaprnlem4N  36163  caragendifcl  39404  fusgreghash2wspv  41499