Theorem 0dif 4402

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 4132	. 2 ⊢ (∅ ∖ 𝐴) ⊆ ∅
2		ss0 4399	. 2 ⊢ ((∅ ∖ 𝐴) ⊆ ∅ → (∅ ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ 𝐴) = ∅

Syntax hints:   = wceq 1542   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324

This theorem is referenced by:  symdif0  5089  fresaun  6763  dffv2  6987  ablfac1eulem  19942  itgioo  25333  newval  27350  nbgr0vtx  28613  imadifxp  31832  sibf0  33333  ballotlemfval0  33494  ballotlemgun  33523  satf0  34363  mdvval  34495  fzdifsuc2  44020  ibliooicc  44687  disjdifb  47494