Theorem dfin2 4219

Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4218. Another version is given by dfin4 4226. (Contributed by NM, 10-Jun-2004.)

Assertion

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

Proof of Theorem dfin2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3438	. . . . . 6 ⊢ 𝑥 ∈ V
2		eldif 3910	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	mpbiran 709	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
4	3	con2bii 357	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
5	4	anbi2i 623	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6		eldif 3910	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
7	5, 6	bitr4i 278	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
8	7	ineqri 4160	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2110  Vcvv 3434   ∖ cdif 3897   ∩ cin 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3436  df-dif 3903  df-in 3907

This theorem is referenced by:  dfun3  4224  dfin3  4225  invdif  4227  difundi  4238  difindi  4240  difdif2  4244