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Theorem 0dif 4335
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 4066 . 2 (∅ ∖ 𝐴) ⊆ ∅
2 ss0 4332 . 2 ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
31, 2ax-mp 5 1 (∅ ∖ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3884  wss 3887  c0 4256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257
This theorem is referenced by:  symdif0  5014  fresaun  6645  dffv2  6863  ablfac1eulem  19675  itgioo  24980  nbgr0vtx  27723  imadifxp  30940  sibf0  32301  ballotlemfval0  32462  ballotlemgun  32491  satf0  33334  mdvval  33466  newval  34039  fzdifsuc2  42849  ibliooicc  43512  disjdifb  46155
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