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Theorem dfin3 4231
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 4101 . 2 (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵))
2 dfun2 4224 . . . 4 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)))
3 ddif 4101 . . . . . 6 (V ∖ (V ∖ 𝐴)) = 𝐴
43difeq1i 4083 . . . . 5 ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵))
54difeq2i 4084 . . . 4 (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
62, 5eqtri 2765 . . 3 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
76difeq2i 4084 . 2 (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵))))
8 dfin2 4225 . 2 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
91, 7, 83eqtr4ri 2776 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3448  cdif 3912  cun 3913  cin 3914
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922
This theorem is referenced by:  difindi  4246
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