Theorem difss2 4083

Description: If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
difss2	⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)

Proof of Theorem difss2

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ (𝐵 ∖ 𝐶))
2		difss 4081	. 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵
3	1, 2	sstrdi 3942	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3894   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-ss 3914

This theorem is referenced by:  difss2d  4084  ssdifsn  4735  sbthlem1  8995  bcthlem2  25247  ismblfin  37701