Theorem 0dif 4290

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 4022	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4287	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1542   ∖ cdif 3840   ⊆ wss 3843  ∅c0 4211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212

This theorem is referenced by:  symdif0  4970  fresaun  6549  dffv2  6763  ablfac1eulem  19313  itgioo  24568  nbgr0vtx  27298  imadifxp  30514  sibf0  31871  ballotlemfval0  32032  ballotlemgun  32061  satf0  32905  mdvval  33037  fzdifsuc2  42387  ibliooicc  43054  disjdifb  45687