Theorem ssdifss 4105

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 4101	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3957	. 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 3913   ⊆ wss 3916

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3919  df-ss 3933

This theorem is referenced by:  ssdifssd  4112  xrsupss  13275  xrinfmss  13276  rpnnen2lem12  16199  lpval  23032  lpdifsn  23036  islp2  23038  lpcls  23257  mblfinlem3  37648  mblfinlem4  37649  voliunnfl  37653  redvmptabs  42343  ssdifcl  43553  sssymdifcl  43554  supxrmnf2  45422  infxrpnf2  45452  fourierdlem102  46199  fourierdlem114  46211  lindslinindimp2lem4  48440  lindslinindsimp2lem5  48441  lindslinindsimp2  48442  lincresunit3  48460