Theorem dfin2 4224

Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4223. Another version is given by dfin4 4231. (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 3445	. . . . . 6 ⊢ 𝑥 ∈ V
2		eldif 3912	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	mpbiran 710	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
4	3	con2bii 357	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
5	4	anbi2i 624	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6		eldif 3912	. . 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 4165	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∖ cdif 3899   ∩ cin 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-dif 3905  df-in 3909

This theorem is referenced by:  dfun3  4229  dfin3  4230  invdif  4232  difundi  4243  difindi  4245  difdif2  4249