Theorem ssdifss 4134

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 4130	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3989	. 2 ⊢ (((𝐴 ∖ 𝐶) ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∖ 𝐶) ⊆ 𝐵)
3	1, 2	mpan 686	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3944   ⊆ wss 3947

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964

This theorem is referenced by:  ssdifssd  4141  xrsupss  13292  xrinfmss  13293  rpnnen2lem12  16172  lpval  22863  lpdifsn  22867  islp2  22869  lpcls  23088  mblfinlem3  36830  mblfinlem4  36831  voliunnfl  36835  ssdifcl  42624  sssymdifcl  42625  supxrmnf2  44441  infxrpnf2  44471  fourierdlem102  45222  fourierdlem114  45234  lindslinindimp2lem4  47229  lindslinindsimp2lem5  47230  lindslinindsimp2  47231  lincresunit3  47249