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

Theorem pwundif 4792
 Description: Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.)
Assertion
Ref Expression
pwundif

Proof of Theorem pwundif
StepHypRef Expression
1 undif1 3903 . 2
2 pwunss 4790 . . . . 5
3 unss 3677 . . . . 5
42, 3mpbir 209 . . . 4
54simpli 458 . . 3
6 ssequn2 3676 . . 3
75, 6mpbi 208 . 2
81, 7eqtr2i 2487 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  =wceq 1395  \cdif 3472  u.cun 3473  C_wss 3475  ~Pcpw 4012 This theorem is referenced by:  pwfilem  7834 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  df-pw 4014
 Copyright terms: Public domain W3C validator