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 4072	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4351	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1569   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-nul 4279

This theorem is used by:  symdif0  5044  fresaun  6754  dffv2  6981  nulchn  18799  chnccat  18806  ablfac1eulem  20294  itgioo  26142  newval  28227  imadifxp  33243  sibf0  35013  ballotlemfval0  35175  ballotlemgun  35204  satf0  36224  mdvval  36356  fzdifsuc2  46420  ibliooicc  47075  disjdifb  50014