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Theorem unvdif 3994
 Description: The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unvdif (𝐴 ∪ (V ∖ 𝐴)) = V

Proof of Theorem unvdif
StepHypRef Expression
1 dfun3 3824 . 2 (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2 disjdif 3992 . . 3 ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
32difeq2i 3687 . 2 (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4 dif0 3904 . 2 (V ∖ ∅) = V
51, 3, 43eqtri 2636 1 (𝐴 ∪ (V ∖ 𝐴)) = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  undif1  3995  dfif4  4051  hashfxnn0  12986  hashfOLD  12988  fullfunfnv  31223  hfext  31460
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