MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssdifeq0 Unicode version

Theorem ssdifeq0 3910
Description: A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
Assertion
Ref Expression
ssdifeq0

Proof of Theorem ssdifeq0
StepHypRef Expression
1 inidm 3706 . . 3
2 ssdifin0 3909 . . 3
31, 2syl5eqr 2512 . 2
4 0ss 3814 . . 3
5 id 22 . . . 4
6 difeq2 3615 . . . 4
75, 6sseq12d 3532 . . 3
84, 7mpbiri 233 . 2
93, 8impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  =wceq 1395  \cdif 3472  i^icin 3474  C_wss 3475   c0 3784
This theorem is referenced by:  disjdifprg  27436  measxun2  28181  measssd  28186
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-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785
  Copyright terms: Public domain W3C validator