Theorem 0dif 4371

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 4102	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4368	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3914   ⊆ wss 3917  ∅c0 4299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-ss 3934  df-nul 4300

This theorem is referenced by:  symdif0  5052  fresaun  6734  dffv2  6959  ablfac1eulem  20011  itgioo  25724  newval  27770  imadifxp  32537  sibf0  34332  ballotlemfval0  34494  ballotlemgun  34523  satf0  35366  mdvval  35498  fzdifsuc2  45315  ibliooicc  45976  disjdifb  48802