Theorem ssdifss 4087

Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Assertion

Ref	Expression
ssdifss	⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Proof of Theorem ssdifss

Step	Hyp	Ref	Expression
1		difss 4083	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3938	. 2 ⊢ (((𝐴 ∖ 𝐶) ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∖ 𝐶) ⊆ 𝐵)
3	1, 2	mpan 690	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3894   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-ss 3914

This theorem is referenced by:  ssdifssd  4094  xrsupss  13208  xrinfmss  13209  rpnnen2lem12  16134  lpval  23054  lpdifsn  23058  islp2  23060  lpcls  23279  mblfinlem3  37709  mblfinlem4  37710  voliunnfl  37714  redvmptabs  42463  ssdifcl  43674  sssymdifcl  43675  supxrmnf2  45541  infxrpnf2  45571  fourierdlem102  46316  fourierdlem114  46328  lindslinindimp2lem4  48572  lindslinindsimp2lem5  48573  lindslinindsimp2  48574  lincresunit3  48592