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Theorem uni0c 3934
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 3933 . 2  |-  ( U. A  =  (/)  <->  A  C_  { (/) } )
2 dfss3 3246 . 2  |-  ( A 
C_  { (/) }  <->  A. x  e.  A  x  e.  {
(/) } )
3 elsn 3731 . . 3  |-  ( x  e.  { (/) }  <->  x  =  (/) )
43ralbii 2643 . 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 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   (/)c0 3531   {csn 3716   U.cuni 3908
This theorem is referenced by:  fin1a2lem13  8128  fctop  16847  cctop  16849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-in 3235  df-ss 3242  df-nul 3532  df-sn 3722  df-uni 3909
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