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Theorem uni0c 3850
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 3849 . 2  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
2 dfss3 3160 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
3 velsn 3624 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
43ralbii 2496 . 2  |-  ( A. x  e.  A  x  e.  { (/) }  <->  A. x  e.  A  x  =  (/) )
51, 2, 43bitri 206 1  |-  ( U. A  =  (/)  <->  A. x  e.  A  x  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468    C_ wss 3144   (/)c0 3437   {csn 3607   U.cuni 3824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-uni 3825
This theorem is referenced by: (None)
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