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Theorem 0dif 4139
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 3899 . 2 (∅ ∖ 𝐴) ⊆ ∅
2 ss0 4136 . 2 ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
31, 2ax-mp 5 1 (∅ ∖ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cdif 3729  wss 3732  c0 4079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-nul 4080
This theorem is referenced by:  symdif0  4753  fresaun  6257  dffv2  6460  ablfac1eulem  18738  itgioo  23873  nbgr0vtx  26531  imadifxp  29862  sibf0  30843  ballotlemfval0  31005  ballotlemgun  31034  mdvval  31849  fzdifsuc2  40163  ibliooicc  40824
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