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Theorem dfun3 4227
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 4221 . 2 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))
2 dfin2 4222 . . . 4 ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵)))
3 ddif 4099 . . . . 5 (V ∖ (V ∖ 𝐵)) = 𝐵
43difeq2i 4082 . . . 4 ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵)
52, 4eqtr2i 2848 . . 3 ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))
65difeq2i 4082 . 2 (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
71, 6eqtri 2847 1 (𝐴𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3480  cdif 3916  cun 3917  cin 3918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926
This theorem is referenced by:  difundi  4241  unvdif  4406
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