Theorem dif0 1756

Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16.

Assertion

Ref	Expression
dif0	⊢ (A ∖ ∅) = A

Proof of Theorem dif0

Step	Hyp	Ref	Expression
1		difid 1755	. . 3 ⊢ (A ∖ A) = ∅
2	1	difeq2i 1585	. 2 ⊢ (A ∖ (A ∖ A)) = (A ∖ ∅)
3		difdif 1595	. 2 ⊢ (A ∖ (A ∖ A)) = A
4	2, 3	eqtr3 1121	1 ⊢ (A ∖ ∅) = A

Syntax hints:   = wceq 1091   ∖ cdif 1484  ∅c0 1707

This theorem is referenced by:  undifv 1760  oe0m0 3128

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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