Theorem ssdifss 4094

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 4090	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3944	. 2 ⊢ (((𝐴 ∖ 𝐶) ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∖ 𝐶) ⊆ 𝐵)
3	1, 2	mpan 691	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3900   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920

This theorem is referenced by:  ssdifssd  4101  xrsupss  13236  xrinfmss  13237  rpnnen2lem12  16162  lpval  23095  lpdifsn  23099  islp2  23101  lpcls  23320  mblfinlem3  37910  mblfinlem4  37911  voliunnfl  37915  redvmptabs  42730  ssdifcl  43927  sssymdifcl  43928  supxrmnf2  45791  infxrpnf2  45821  fourierdlem102  46566  fourierdlem114  46578  lindslinindimp2lem4  48821  lindslinindsimp2lem5  48822  lindslinindsimp2  48823  lincresunit3  48841