Theorem dfin2 4194

Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4193. Another version is given by dfin4 4201. (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 3436	. . . . . 6 ⊢ 𝑥 ∈ V
2		eldif 3897	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵))
3	1, 2	mpbiran 706	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
4	3	con2bii 358	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
5	4	anbi2i 623	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
6		eldif 3897	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
7	5, 6	bitr4i 277	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
8	7	ineqri 4138	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894

This theorem is referenced by:  dfun3  4199  dfin3  4200  invdif  4202  difundi  4213  difindi  4215  difdif2  4220