Theorem dfun3 1671

Description: Union defined in terms of intersection (DeMorgan's law). Definition of union in [Mendelson] p. 231.

Assertion

Ref	Expression
dfun3	⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∩ (V ∖ B)))

Proof of Theorem dfun3

Step	Hyp	Ref	Expression
1		dfun2 1668	. 2 ⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∖ B)
2		dfin2 1669	. . . 4 ⊢ ((V ∖ A) ∩ (V ∖ B)) = ((V ∖ A) ∖ (V ∖ (V ∖ B)))
3		ddif 1597	. . . . 5 ⊢ (V ∖ (V ∖ B)) = B
4	3	difeq2i 1585	. . . 4 ⊢ ((V ∖ A) ∖ (V ∖ (V ∖ B))) = ((V ∖ A) ∖ B)
5	2, 4	eqtr2 1120	. . 3 ⊢ ((V ∖ A) ∖ B) = ((V ∖ A) ∩ (V ∖ B))
6	5	difeq2i 1585	. 2 ⊢ (V ∖ ((V ∖ A) ∖ B)) = (V ∖ ((V ∖ A) ∩ (V ∖ B)))
7	1, 6	eqtr 1119	1 ⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∩ (V ∖ B)))

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

This theorem is referenced by:  difundi 1681  undifv 1760

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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