Theorem ssdifin0 3632

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 3481	. 2 ⊢ (A ⊆ (B ∖ C) → (A ∩ C) ⊆ ((B ∖ C) ∩ C))
2		incom 3449	. . 3 ⊢ ((B ∖ C) ∩ C) = (C ∩ (B ∖ C))
3		disjdif 3623	. . 3 ⊢ (C ∩ (B ∖ C)) = ∅
4	2, 3	eqtri 2373	. 2 ⊢ ((B ∖ C) ∩ C) = ∅
5		sseq0 3583	. 2 ⊢ (((A ∩ C) ⊆ ((B ∖ C) ∩ C) ∧ ((B ∖ C) ∩ C) = ∅) → (A ∩ C) = ∅)
6	1, 4, 5	sylancl 643	1 ⊢ (A ⊆ (B ∖ C) → (A ∩ C) = ∅)

Syntax hints:   → wi 4   = wceq 1642   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258  ∅c0 3551

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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  ssdifeq0  3633