Theorem 0dif 4356

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 4087	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4353	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1559   ∖ 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 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3905  df-ss 3919  df-nul 4284

This theorem is referenced by:  symdif0  5039  fresaun  6729  dffv2  6956  nulchn  18641  chnccat  18648  ablfac1eulem  20104  itgioo  25865  newval  27915  imadifxp  32760  sibf0  34591  ballotlemfval0  34753  ballotlemgun  34782  satf0  35682  mdvval  35814  fzdifsuc2  45849  ibliooicc  46505  disjdifb  49391