Theorem dfin3 4197

Description: Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)

Assertion

Ref	Expression
dfin3	⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Proof of Theorem dfin3

Step	Hyp	Ref	Expression
1		ddif 4067	. 2 ⊢ (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵))
2		dfun2 4190	. . . 4 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)))
3		ddif 4067	. . . . . 6 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴
4	3	difeq1i 4049	. . . . 5 ⊢ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵))
5	4	difeq2i 4050	. . . 4 ⊢ (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
6	2, 5	eqtri 2766	. . 3 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
7	6	difeq2i 4050	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵))))
8		dfin2 4191	. 2 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵))
9	1, 7, 8	3eqtr4ri 2777	1 ⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Syntax hints:   = wceq 1539  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890

This theorem is referenced by:  difindi  4212