Theorem difss2 4109

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 4107	. 2 ⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵
3	1, 2	sstrdi 3978	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∖ cdif 3932   ⊆ wss 3935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951

This theorem is referenced by:  difss2d  4110  ssdifsn  4713  sbthlem1  8621  bcthlem2  23922  ismblfin  34927