Theorem ssdifin0 3909

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

Proof of Theorem ssdifin0

Step	Hyp	Ref	Expression
1		ssrin 3722	. 2
2		incom 3690	. . 3
3		disjdif 3900	. . 3
4	2, 3	eqtri 2486	. 2
5		sseq0 3817	. 2
6	1, 4, 5	sylancl 662	1

Syntax hints:  ->wi 4  =wceq 1395  \cdif 3472  i^icin 3474  C_wss 3475  c0 3784

This theorem is referenced by:  ssdifeq0  3910  marypha1lem  7913  numacn  8451  mreexexlem2d  15042  mreexexlem4d  15044  nrmsep2  19857  isnrm3  19860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785