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Theorem unvdif 4019
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 3846 . 2 (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2 disjdif 4017 . . 3 ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
32difeq2i 3708 . 2 (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4 dif0 3929 . 2 (V ∖ ∅) = V
51, 3, 43eqtri 2647 1 (𝐴 ∪ (V ∖ 𝐴)) = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  Vcvv 3189  cdif 3556  cun 3557  cin 3558  c0 3896
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 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897
This theorem is referenced by:  undif1  4020  dfif4  4078  hashfxnn0  13071  hashfOLD  13073  fullfunfnv  31722  hfext  31959
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