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Theorem 0dif 3691
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 3466 . 2  |-  ( (/)  \  A )  C_  (/)
2 ss0 3650 . 2  |-  ( (
(/)  \  A )  C_  (/)  ->  ( (/)  \  A
)  =  (/) )
31, 2ax-mp 8 1  |-  ( (/)  \  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3309    C_ wss 3312   (/)c0 3620
This theorem is referenced by:  fresaun  5605  dffv2  5787  ablfac1eulem  15618  bwth  17461  itgioo  19695  imadifxp  24026  sibf0  24637  ballotlemfval0  24741  ballotlemgun  24770  symdif0  25623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621
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