Theorem 0dif 4411

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 4146	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4408	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1537   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340

This theorem is referenced by:  symdif0  5090  fresaun  6780  dffv2  7004  ablfac1eulem  20107  itgioo  25866  newval  27909  imadifxp  32621  sibf0  34316  ballotlemfval0  34477  ballotlemgun  34506  satf0  35357  mdvval  35489  fzdifsuc2  45261  ibliooicc  45927  disjdifb  48658