Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssdifin0 Unicode version

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
StepHypRef Expression
1 ssrin 3722 . 2
2 incom 3690 . . 3
3 disjdif 3900 . . 3
42, 3eqtri 2486 . 2
5 sseq0 3817 . 2
61, 4, 5sylancl 662 1
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator