Theorem 0dif 4352

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 4083	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4349	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1541   ∖ cdif 3894   ⊆ wss 3897  ∅c0 4280

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 3900  df-ss 3914  df-nul 4281

This theorem is referenced by:  symdif0  5031  fresaun  6694  dffv2  6917  nulchn  18525  chnccat  18532  ablfac1eulem  19986  itgioo  25744  newval  27796  imadifxp  32581  sibf0  34347  ballotlemfval0  34509  ballotlemgun  34538  satf0  35416  mdvval  35548  fzdifsuc2  45421  ibliooicc  46079  disjdifb  48920