Theorem dif0 3621

Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
dif0	⊢ (A ∖ ∅) = A

Proof of Theorem dif0

Step	Hyp	Ref	Expression
1		difid 3619	. . 3 ⊢ (A ∖ A) = ∅
2	1	difeq2i 3383	. 2 ⊢ (A ∖ (A ∖ A)) = (A ∖ ∅)
3		difdif 3393	. 2 ⊢ (A ∖ (A ∖ A)) = A
4	2, 3	eqtr3i 2375	1 ⊢ (A ∖ ∅) = A

Syntax hints:   = wceq 1642   ∖ cdif 3207  ∅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-ne 2519  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:  undifv  3625