Theorem 0dif 4355

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

Step	Hyp	Ref	Expression
1		difss 4086	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4352	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1541   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3905  df-ss 3919  df-nul 4284

This theorem is referenced by:  symdif0  5033  fresaun  6694  dffv2  6917  nulchn  18525  chnccat  18532  ablfac1eulem  19987  itgioo  25745  newval  27797  imadifxp  32579  sibf0  34345  ballotlemfval0  34507  ballotlemgun  34536  satf0  35414  mdvval  35546  fzdifsuc2  45357  ibliooicc  46015  disjdifb  48847