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Theorem dfin3 4283
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 4151 . 2 (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵))
2 dfun2 4276 . . . 4 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)))
3 ddif 4151 . . . . . 6 (V ∖ (V ∖ 𝐴)) = 𝐴
43difeq1i 4132 . . . . 5 ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵))
54difeq2i 4133 . . . 4 (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
62, 5eqtri 2763 . . 3 ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵)))
76difeq2i 4133 . 2 (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵))))
8 dfin2 4277 . 2 (𝐴𝐵) = (𝐴 ∖ (V ∖ 𝐵))
91, 7, 83eqtr4ri 2774 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cdif 3960  cun 3961  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970
This theorem is referenced by:  difindi  4298
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