Theorem 0dif 4309

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 4059	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4306	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1538   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244

This theorem is referenced by:  symdif0  4970  fresaun  6523  dffv2  6733  ablfac1eulem  19187  itgioo  24419  nbgr0vtx  27146  imadifxp  30364  sibf0  31702  ballotlemfval0  31863  ballotlemgun  31892  satf0  32732  mdvval  32864  fzdifsuc2  41942  ibliooicc  42613