Theorem 0dif 4362

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 4092	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4359	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1563   ∖ cdif 3904   ⊆ wss 3907  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3910  df-ss 3924  df-nul 4289

This theorem is referenced by:  symdif0  5046  fresaun  6739  dffv2  6966  nulchn  18663  chnccat  18670  ablfac1eulem  20132  itgioo  25932  newval  27982  imadifxp  32852  sibf0  34636  ballotlemfval0  34798  ballotlemgun  34827  satf0  35730  mdvval  35862  fzdifsuc2  45888  ibliooicc  46544  disjdifb  49440