Theorem difundi 1681

Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29.

Assertion

Ref	Expression
difundi	⊢ (A ∖ (B ∪ C)) = ((A ∖ B) ∩ (A ∖ C))

Proof of Theorem difundi

Step	Hyp	Ref	Expression
1		dfun3 1671	. . 3 ⊢ (B ∪ C) = (V ∖ ((V ∖ B) ∩ (V ∖ C)))
2	1	difeq2i 1585	. 2 ⊢ (A ∖ (B ∪ C)) = (A ∖ (V ∖ ((V ∖ B) ∩ (V ∖ C))))
3		inindi 1654	. . 3 ⊢ (A ∩ ((V ∖ B) ∩ (V ∖ C))) = ((A ∩ (V ∖ B)) ∩ (A ∩ (V ∖ C)))
4		dfin2 1669	. . 3 ⊢ (A ∩ ((V ∖ B) ∩ (V ∖ C))) = (A ∖ (V ∖ ((V ∖ B) ∩ (V ∖ C))))
5		invdif 1674	. . . 4 ⊢ (A ∩ (V ∖ B)) = (A ∖ B)
6		invdif 1674	. . . 4 ⊢ (A ∩ (V ∖ C)) = (A ∖ C)
7	5, 6	ineq12i 1643	. . 3 ⊢ ((A ∩ (V ∖ B)) ∩ (A ∩ (V ∖ C))) = ((A ∖ B) ∩ (A ∖ C))
8	3, 4, 7	3eqtr3 1124	. 2 ⊢ (A ∖ (V ∖ ((V ∖ B) ∩ (V ∖ C)))) = ((A ∖ B) ∩ (A ∖ C))
9	2, 8	eqtr 1119	1 ⊢ (A ∖ (B ∪ C)) = ((A ∖ B) ∩ (A ∖ C))

Syntax hints:   = wceq 1091  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486

This theorem is referenced by:  undm 1685

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-un 1490  df-in 1491

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