MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0dif Structured version   Unicode version

Theorem 0dif 3885
Description: The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
0dif  |-  ( (/)  \  A )  =  (/)

Proof of Theorem 0dif
StepHypRef Expression
1 difss 3616 . 2  |-  ( (/)  \  A )  C_  (/)
2 ss0 3802 . 2  |-  ( (
(/)  \  A )  C_  (/)  ->  ( (/)  \  A
)  =  (/) )
31, 2ax-mp 5 1  |-  ( (/)  \  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383    \ cdif 3458    C_ wss 3461   (/)c0 3770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3771
This theorem is referenced by:  fresaun  5746  dffv2  5931  ablfac1eulem  16997  bwthOLD  19784  itgioo  22095  imadifxp  27330  sibf0  28149  ballotlemfval0  28307  ballotlemgun  28336  mdvval  28737  symdif0  29449  ibliooicc  31660
  Copyright terms: Public domain W3C validator