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Theorem 0dif 4120
 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 3880 . 2 (∅ ∖ 𝐴) ⊆ ∅
2 ss0 4117 . 2 ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
31, 2ax-mp 5 1 (∅ ∖ 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∖ cdif 3712   ⊆ wss 3715  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059 This theorem is referenced by:  symdif0  4749  fresaun  6236  dffv2  6434  ablfac1eulem  18691  itgioo  23801  nbgr0vtx  26472  imadifxp  29742  sibf0  30726  ballotlemfval0  30887  ballotlemgun  30916  mdvval  31729  fzdifsuc2  40041  ibliooicc  40708
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