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 4083	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4352	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279

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 2147  ax-9 2155  ax-ext 2733

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 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-ss 3916  df-nul 4280

This theorem is used by:  symdif0  5045  fresaun  6745  dffv2  6972  nulchn  18773  chnccat  18780  ablfac1eulem  20268  itgioo  26116  newval  28203  imadifxp  33177  sibf0  34949  ballotlemfval0  35111  ballotlemgun  35140  satf0  36106  mdvval  36238  fzdifsuc2  46269  ibliooicc  46925  disjdifb  49864