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

Theorem unvdif 3902
 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 3735 . 2
2 disjdif 3900 . . 3
32difeq2i 3618 . 2
4 dif0 3898 . 2
51, 3, 43eqtri 2490 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395   cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474   c0 3784 This theorem is referenced by:  undif1  3903  dfif4  3956  hashf  12412  fullfunfnv  29596  hfext  29840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785
 Copyright terms: Public domain W3C validator