Theorem dfin2 1669

Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 1668. Another version is given by dfin4 1673.

Assertion

Ref	Expression
dfin2	⊢ (A ∩ B) = (A ∖ (V ∖ B))

Proof of Theorem dfin2

Step	Hyp	Ref	Expression
1		eldif 1496	. . . . . 6 ⊢ (x ∈ (V ∖ B) ↔ (x ∈ V ∧ ¬ x ∈ B))
2		visset 1350	. . . . . 6 ⊢ x ∈ V
3	1, 2	mpbiran 547	. . . . 5 ⊢ (x ∈ (V ∖ B) ↔ ¬ x ∈ B)
4	3	bicon2i 194	. . . 4 ⊢ (x ∈ B ↔ ¬ x ∈ (V ∖ B))
5	4	anbi2i 367	. . 3 ⊢ ((x ∈ A ∧ x ∈ B) ↔ (x ∈ A ∧ ¬ x ∈ (V ∖ B)))
6		elin 1635	. . 3 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
7		eldif 1496	. . 3 ⊢ (x ∈ (A ∖ (V ∖ B)) ↔ (x ∈ A ∧ ¬ x ∈ (V ∖ B)))
8	5, 6, 7	3bitr4 158	. 2 ⊢ (x ∈ (A ∩ B) ↔ x ∈ (A ∖ (V ∖ B)))
9	8	cleqri 1101	1 ⊢ (A ∩ B) = (A ∖ (V ∖ B))

Syntax hints:  ¬ wn 1   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∩ cin 1486

This theorem is referenced by:  dfun3 1671  dfin3 1672  invdif 1674  difundi 1681  difindi 1683

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-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

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