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Theorem 0dif 4358
 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 4111 . 2 (∅ ∖ 𝐴) ⊆ ∅
2 ss0 4355 . 2 ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
31, 2ax-mp 5 1 (∅ ∖ 𝐴) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   ∖ cdif 3936   ⊆ wss 3939  ∅c0 4294 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-in 3946  df-ss 3955  df-nul 4295 This theorem is referenced by:  symdif0  5003  fresaun  6545  dffv2  6752  ablfac1eulem  19116  itgioo  24331  nbgr0vtx  27052  imadifxp  30266  sibf0  31478  ballotlemfval0  31639  ballotlemgun  31668  satf0  32503  mdvval  32635  fzdifsuc2  41438  ibliooicc  42117
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