Theorem dfun2 1668

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 1669 and dfss4 1667 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation ∖ (class difference).

Assertion

Ref	Expression
dfun2	⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∖ B))

Proof of Theorem dfun2

Step	Hyp	Ref	Expression
1		eldif 1496	. . . . . . 7 ⊢ (x ∈ (V ∖ A) ↔ (x ∈ V ∧ ¬ x ∈ A))
2		visset 1350	. . . . . . 7 ⊢ x ∈ V
3	1, 2	mpbiran 547	. . . . . 6 ⊢ (x ∈ (V ∖ A) ↔ ¬ x ∈ A)
4	3	anbi1i 368	. . . . 5 ⊢ ((x ∈ (V ∖ A) ∧ ¬ x ∈ B) ↔ (¬ x ∈ A ∧ ¬ x ∈ B))
5		eldif 1496	. . . . 5 ⊢ (x ∈ ((V ∖ A) ∖ B) ↔ (x ∈ (V ∖ A) ∧ ¬ x ∈ B))
6		ioran 254	. . . . 5 ⊢ (¬ (x ∈ A ∨ x ∈ B) ↔ (¬ x ∈ A ∧ ¬ x ∈ B))
7	4, 5, 6	3bitr4 158	. . . 4 ⊢ (x ∈ ((V ∖ A) ∖ B) ↔ ¬ (x ∈ A ∨ x ∈ B))
8	7	bicon2i 194	. . 3 ⊢ ((x ∈ A ∨ x ∈ B) ↔ ¬ x ∈ ((V ∖ A) ∖ B))
9		elun 1601	. . 3 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
10		eldif 1496	. . . 4 ⊢ (x ∈ (V ∖ ((V ∖ A) ∖ B)) ↔ (x ∈ V ∧ ¬ x ∈ ((V ∖ A) ∖ B)))
11	10, 2	mpbiran 547	. . 3 ⊢ (x ∈ (V ∖ ((V ∖ A) ∖ B)) ↔ ¬ x ∈ ((V ∖ A) ∖ B))
12	8, 9, 11	3bitr4 158	. 2 ⊢ (x ∈ (A ∪ B) ↔ x ∈ (V ∖ ((V ∖ A) ∖ B)))
13	12	cleqri 1101	1 ⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∖ B))

Syntax hints:  ¬ wn 1   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485

This theorem is referenced by:  dfun3 1671  dfin3 1672

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

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