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Theorem uni0c 3853
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 3852 . 2  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
2 dfss3 3170 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
3 elsn 3655 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
43ralbii 2567 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
51, 2, 43bitri 262 1  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827
This theorem is referenced by:  fin1a2lem13  8038  fctop  16741  cctop  16743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-uni 3828
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