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

Step	Hyp	Ref	Expression
1		undif1 3903	. 2
2		pwunss 4790	. . . . 5
3		unss 3677	. . . . 5
4	2, 3	mpbir 209	. . . 4
5	4	simpli 458	. . 3
6		ssequn2 3676	. . 3
7	5, 6	mpbi 208	. 2
8	1, 7	eqtr2i 2487	1

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