Theorem dfin2 4223

Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 4222. Another version is given by dfin4 4230. (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		velcomp 3919	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
2	1	con2bii 360	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵))
3	2	anbi2i 634	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
4		eldif 3914	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵)))
5	3, 4	bitr4i 281	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)))
6	5	ineqri 4164	1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ∖ cdif 3901   ∩ cin 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-in 3911

This theorem is used by:  dfun3  4228  dfin3  4229  invdif  4231  difundi  4242  difindi  4244  difdif2  4248