Theorem ssdifd 4168

Description: If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4167. (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 4167	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

Syntax hints:   → wi 4   ∖ cdif 3973   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-ss 3993

This theorem is referenced by:  ssdif2d  4171  domunsncan  9138  fin1a2lem13  10481  seqcoll2  14514  rpnnen2lem11  16272  coprmprod  16708  mrieqv2d  17697  mrissmrid  17699  mreexexlem4d  17705  acsfiindd  18623  subdrgint  20826  lsppratlem3  21174  lsppratlem4  21175  f1lindf  21865  lpss3  23173  lpcls  23393  fin1aufil  23961  rrxmval  25458  rrxmetlem  25460  uniioombllem3  25639  i1fmul  25750  itg1addlem4  25753  itg1addlem4OLD  25754  itg1climres  25769  limciun  25949  ig1peu  26234  ig1pdvds  26239  fusgreghash2wspv  30367  chnind  32983  pmtrcnel2  33083  pmtrcnelor  33084  tocyccntz  33137  elrspunidl  33421  elrspunsn  33422  indsumin  33986  sitgclg  34307  mthmpps  35550  poimirlem11  37591  poimirlem12  37592  poimirlem15  37595  dochfln0  41434  lcfl6  41457  lcfrlem16  41515  hdmaprnlem4N  41810  tfsconcatlem  43298  caragendifcl  46435