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Theorem un00 3862
 Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
un00

Proof of Theorem un00
StepHypRef Expression
1 uneq12 3652 . . 3
2 un0 3810 . . 3
31, 2syl6eq 2514 . 2
4 ssun1 3666 . . . . 5
5 sseq2 3525 . . . . 5
64, 5mpbii 211 . . . 4
7 ss0b 3815 . . . 4
86, 7sylib 196 . . 3
9 ssun2 3667 . . . . 5
10 sseq2 3525 . . . . 5
119, 10mpbii 211 . . . 4
12 ss0b 3815 . . . 4
1311, 12sylib 196 . . 3
148, 13jca 532 . 2
153, 14impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  u.cun 3473  C_wss 3475   c0 3784 This theorem is referenced by:  undisj1  3878  undisj2  3879  disjpr2  4092  rankxplim3  8320  ssxr  9675  rpnnen2  13959  wwlknext  24724  asindmre  30102 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-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785
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