Theorem uni0c 4275

Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Assertion

Ref	Expression
uni0c

Distinct variable group:   ,

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 4274	. 2
2		dfss3 3493	. 2
3		elsn 4043	. . 3
4	3	ralbii 2888	. 2
5	1, 2, 4	3bitri 271	1

Syntax hints:  <->wb 184  =wceq 1395  e.wcel 1818  A.wral 2807  C_wss 3475  c0 3784  {csn 4029  U.cuni 4249

This theorem is referenced by:  fin1a2lem13  8813  fctop  19505  cctop  19507

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-rex 2813  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250