Theorem ssdifin0 3496

Description: A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
ssdifin0	⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅)

Proof of Theorem ssdifin0

Step	Hyp	Ref	Expression
1		ssrin 3352	. 2 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶))
2		incom 3319	. . 3 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = (𝐶 ∩ (𝐵 ∖ 𝐶))
3		disjdif 3487	. . 3 ⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅
4	2, 3	eqtri 2191	. 2 ⊢ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅
5		sseq0 3456	. 2 ⊢ (((𝐴 ∩ 𝐶) ⊆ ((𝐵 ∖ 𝐶) ∩ 𝐶) ∧ ((𝐵 ∖ 𝐶) ∩ 𝐶) = ∅) → (𝐴 ∩ 𝐶) = ∅)
6	1, 4, 5	sylancl 411	1 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐶) → (𝐴 ∩ 𝐶) = ∅)

Syntax hints:   → wi 4   = wceq 1348   ∖ cdif 3118   ∩ cin 3120   ⊆ wss 3121  ∅c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  ssdifeq0  3497