Theorem 0dif 4355

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 4108	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4352	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1537   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292

This theorem is referenced by:  symdif0  5007  fresaun  6549  dffv2  6756  ablfac1eulem  19194  itgioo  24416  nbgr0vtx  27138  imadifxp  30351  sibf0  31592  ballotlemfval0  31753  ballotlemgun  31782  satf0  32619  mdvval  32751  fzdifsuc2  41626  ibliooicc  42305