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Theorem dfun3 3819
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 3816 . 2 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
2 dfin2 3817 . . . 4 ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵)))
3 ddif 3699 . . . . 5 (V ∖ (V ∖ 𝐵)) = 𝐵
43difeq2i 3682 . . . 4 ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵)
52, 4eqtr2i 2628 . . 3 ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
65difeq2i 3682 . 2 (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
71, 6eqtri 2627 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  Vcvv 3168  cdif 3532  cun 3533  cin 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542
This theorem is referenced by:  difundi  3833  unvdif  3989
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