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Theorem dfin3 3825
 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 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Proof of Theorem dfin3
StepHypRef Expression
1 ddif 3704 . 2 (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵))
2 dfun2 3821 . . . 4 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)))
3 ddif 3704 . . . . . 6 (V ∖ (V ∖ 𝐴)) = 𝐴
43difeq1i 3686 . . . . 5 ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵))
54difeq2i 3687 . . . 4 (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
62, 5eqtri 2632 . . 3 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
76difeq2i 3687 . 2 (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵))))
8 dfin2 3822 . 2 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
91, 7, 83eqtr4ri 2643 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547 This theorem is referenced by:  difindi  3840
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