Theorem 0dif 4405

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 4136	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4402	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334

This theorem is referenced by:  symdif0  5085  fresaun  6779  dffv2  7004  ablfac1eulem  20092  itgioo  25851  newval  27894  imadifxp  32614  sibf0  34336  ballotlemfval0  34498  ballotlemgun  34527  satf0  35377  mdvval  35509  fzdifsuc2  45322  ibliooicc  45986  disjdifb  48729