Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unvdif Structured version   Unicode version

Theorem unvdif 3866
 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

Proof of Theorem unvdif
StepHypRef Expression
1 dfun3 3708 . 2
2 disjdif 3864 . . 3
32difeq2i 3577 . 2
4 dif0 3862 . 2
51, 3, 43eqtri 2453 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437  cvv 3078   cdif 3430   cun 3431   cin 3432  c0 3758 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759 This theorem is referenced by:  undif1  3867  dfif4  3921  hashf  12515  fullfunfnv  30699  hfext  30936
 Copyright terms: Public domain W3C validator