Theorem ssdifss 4066

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 4062	. 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴
2		sstr 3925	. 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 3880   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900

This theorem is referenced by:  ssdifssd  4073  xrsupss  12972  xrinfmss  12973  rpnnen2lem12  15862  lpval  22198  lpdifsn  22202  islp2  22204  lpcls  22423  mblfinlem3  35743  mblfinlem4  35744  voliunnfl  35748  ssdifcl  41067  sssymdifcl  41068  supxrmnf2  42863  infxrpnf2  42893  fourierdlem102  43639  fourierdlem114  43651  lindslinindimp2lem4  45690  lindslinindsimp2lem5  45691  lindslinindsimp2  45692  lincresunit3  45710