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Theorem unvdif 4184
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 4014 . 2 (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2 disjdif 4182 . . 3 ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
32difeq2i 3876 . 2 (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4 dif0 4097 . 2 (V ∖ ∅) = V
51, 3, 43eqtri 2797 1 (𝐴 ∪ (V ∖ 𝐴)) = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  Vcvv 3351  cdif 3720  cun 3721  cin 3722  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064
This theorem is referenced by:  undif1  4185  dfif4  4240  hashfxnn0  13328  hashfOLD  13330  fullfunfnv  32390  hfext  32627
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