Theorem ssdifss 4140

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 4136	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3992	. 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 3948   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968

This theorem is referenced by:  ssdifssd  4147  xrsupss  13351  xrinfmss  13352  rpnnen2lem12  16261  lpval  23147  lpdifsn  23151  islp2  23153  lpcls  23372  mblfinlem3  37666  mblfinlem4  37667  voliunnfl  37671  redvmptabs  42390  ssdifcl  43584  sssymdifcl  43585  supxrmnf2  45444  infxrpnf2  45474  fourierdlem102  46223  fourierdlem114  46235  lindslinindimp2lem4  48378  lindslinindsimp2lem5  48379  lindslinindsimp2  48380  lincresunit3  48398