Theorem 0dif 4366

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 4093	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4362	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3905   ⊆ wss 3908  ∅c0 4289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-ss 3925  df-nul 4290

This theorem is used by:  symdif0  5056  fresaun  6756  dffv2  6983  nulchn  18700  chnccat  18707  ablfac1eulem  20175  itgioo  26012  newval  28065  imadifxp  32983  sibf0  34756  ballotlemfval0  34918  ballotlemgun  34947  satf0  35885  mdvval  36017  fzdifsuc2  46070  ibliooicc  46726  disjdifb  49629