Theorem 0dif 4357

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 4088	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4354	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1541   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285

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 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-ss 3918  df-nul 4286

This theorem is referenced by:  symdif0  5040  fresaun  6705  dffv2  6929  nulchn  18542  chnccat  18549  ablfac1eulem  20003  itgioo  25773  newval  27831  imadifxp  32676  sibf0  34491  ballotlemfval0  34653  ballotlemgun  34682  satf0  35566  mdvval  35698  fzdifsuc2  45558  ibliooicc  46215  disjdifb  49055