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Theorem 0dif 3929
 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 3699 . 2 (∅ ∖ 𝐴) ⊆ ∅
2 ss0 3926 . 2 ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
31, 2ax-mp 5 1 (∅ ∖ 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  symdif0  4533  fresaun  5988  dffv2  6181  ablfac1eulem  18294  itgioo  23388  imadifxp  28796  sibf0  29723  ballotlemfval0  29884  ballotlemgun  29913  mdvval  30655  fzdifsuc2  38466  ibliooicc  38863  nbgr0vtx  40578
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