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Theorem unvdif 4475
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 4276 . 2 (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2 disjdif 4472 . . 3 ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
32difeq2i 4123 . 2 (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4 dif0 4378 . 2 (V ∖ ∅) = V
51, 3, 43eqtri 2769 1 (𝐴 ∪ (V ∖ 𝐴)) = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cdif 3948  cun 3949  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334
This theorem is referenced by:  undif1  4476  dfif4  4541  hashfxnn0  14376  fullfunfnv  35947  hfext  36184
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