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Theorem dfun3 3847
Description: Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
Assertion
Ref Expression
dfun3 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Proof of Theorem dfun3
StepHypRef Expression
1 dfun2 3843 . 2 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
2 dfin2 3844 . . . 4 ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵)))
3 ddif 3726 . . . . 5 (V ∖ (V ∖ 𝐵)) = 𝐵
43difeq2i 3709 . . . 4 ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵)
52, 4eqtr2i 2644 . . 3 ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
65difeq2i 3709 . 2 (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
71, 6eqtri 2643 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3190  cdif 3557  cun 3558  cin 3559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567
This theorem is referenced by:  difundi  3861  unvdif  4020
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