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Theorem uni0c 3679
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 3678 . 2  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
2 dfss3 3015 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
3 velsn 3463 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
43ralbii 2384 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
51, 2, 43bitri 204 1  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2999   (/)c0 3286   {csn 3446   U.cuni 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3452  df-uni 3654
This theorem is referenced by: (None)
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