Theorem ssdifd 4084

Description: If 𝐴 is contained in 𝐵, then (𝐴 ∖ 𝐶) is contained in (𝐵 ∖ 𝐶). Deduction form of ssdif 4083. (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 4083	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   ∖ cdif 4048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050

This theorem is used by:  ssdif2d  4087  domunsncan  9134  fin1a2lem13  10529  seqcoll2  14651  rpnnen2lem11  16436  coprmprod  16880  mrieqv2d  17863  mrissmrid  17865  mreexexlem4d  17871  acsfiindd  18777  chnind  18845  chnrev  18851  subdrgint  21111  lsppratlem3  21478  lsppratlem4  21479  f1lindf  22183  lpss3  23517  lpcls  23737  fin1aufil  24306  rrxmval  25781  rrxmetlem  25783  uniioombllem3  25961  i1fmul  26072  itg1addlem4  26075  itg1climres  26090  limciun  26269  ig1peu  26548  ig1pdvds  26553  fusgreghash2wspv  30993  indsumin  33524  pmtrcnel2  33750  pmtrcnelor  33751  tocyccntz  33804  edompidlem  33960  elrspunidl  34082  elrspunsn  34083  sitgclg  35085  mthmpps  36555  poimirlem11  38817  poimirlem12  38818  poimirlem15  38821  dochfln0  42800  lcfl6  42823  lcfrlem16  42881  hdmaprnlem4N  43176  tfsconcatlem  44599  caragendifcl  47797