Theorem ssdifss 4136

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 4132	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3991	. 2 ⊢ (((𝐴 ∖ 𝐶) ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∖ 𝐶) ⊆ 𝐵)
3	1, 2	mpan 689	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3946   ⊆ wss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966

This theorem is referenced by:  ssdifssd  4143  xrsupss  13288  xrinfmss  13289  rpnnen2lem12  16168  lpval  22643  lpdifsn  22647  islp2  22649  lpcls  22868  mblfinlem3  36527  mblfinlem4  36528  voliunnfl  36532  ssdifcl  42322  sssymdifcl  42323  supxrmnf2  44143  infxrpnf2  44173  fourierdlem102  44924  fourierdlem114  44936  lindslinindimp2lem4  47142  lindslinindsimp2lem5  47143  lindslinindsimp2  47144  lincresunit3  47162