Theorem 0dif 3622

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 3394	. 2 ⊢ (∅ ∖ A) ⊆ ∅
2		ss0 3582	. 2 ⊢ ((∅ ∖ A) ⊆ ∅ → (∅ ∖ A) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (∅ ∖ A) = ∅

Syntax hints:   = wceq 1642   ∖ cdif 3207   ⊆ wss 3258  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)