Theorem ssdifss 3398

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 3394	. 2 ⊢ (A ∖ C) ⊆ A
2		sstr 3281	. 2 ⊢ (((A ∖ C) ⊆ A ∧ A ⊆ B) → (A ∖ C) ⊆ B)
3	1, 2	mpan 651	1 ⊢ (A ⊆ B → (A ∖ C) ⊆ B)

Syntax hints:   → wi 4   ∖ cdif 3207   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by:  ssdifssd  3405