Theorem 0dif 4401

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 4131	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4398	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323

This theorem is referenced by:  symdif0  5088  fresaun  6762  dffv2  6986  ablfac1eulem  19990  itgioo  25665  newval  27695  nbgr0vtx  29046  imadifxp  32265  sibf0  33797  ballotlemfval0  33958  ballotlemgun  33987  satf0  34827  mdvval  34959  fzdifsuc2  44479  ibliooicc  45146  disjdifb  47656