Theorem 0dif 4359

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 4075	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4355	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282

This theorem is used by:  symdif0  5048  fresaun  6778  dffv2  7006  nulchn  18825  chnccat  18832  ablfac1eulem  20321  itgioo  26169  newval  28254  imadifxp  33270  sibf0  35049  ballotlemfval0  35211  ballotlemgun  35240  satf0  36317  mdvval  36449  fzdifsuc2  46541  ibliooicc  47196  disjdifb  50135