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Theorem uni0c 4043
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  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
Distinct variable group:    x, A

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 4042 . 2  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
2 dfss3 3340 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
3 elsn 3831 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
43ralbii 2731 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
51, 2, 43bitri 264 1  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   (/)c0 3630   {csn 3816   U.cuni 4017
This theorem is referenced by:  fin1a2lem13  8297  fctop  17073  cctop  17075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-uni 4018
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