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

Theorem 0dif 3903
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 3636 . 2  |-  ( (/)  \  A )  C_  (/)
2 ss0 3821 . 2  |-  ( (
(/)  \  A )  C_  (/)  ->  ( (/)  \  A
)  =  (/) )
31, 2ax-mp 5 1  |-  ( (/)  \  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3478    C_ wss 3481   (/)c0 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-dif 3484  df-in 3488  df-ss 3495  df-nul 3791
This theorem is referenced by:  fresaun  5761  dffv2  5946  ablfac1eulem  16972  bwthOLD  19756  itgioo  22067  imadifxp  27249  sibf0  28069  ballotlemfval0  28227  ballotlemgun  28256  mdvval  28657  symdif0  29369  ibliooicc  31580
  Copyright terms: Public domain W3C validator