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Theorem uni0c 3597
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.  (/)  (/)
Distinct variable group:   ,

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 3596 . 2  U.  (/)  C_  {
(/) }
2 dfss3 2929 . 2 
C_  { (/) }  { (/)
}
3 elsn 3382 . . 3  { (/) }  (/)
43ralbii 2324 . 2  { (/) }  (/)
51, 2, 43bitri 195 1  U.  (/)  (/)
Colors of variables: wff set class
Syntax hints:   wb 98   wceq 1242   wcel 1390  wral 2300    C_ wss 2911   (/)c0 3218   {csn 3367   U.cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-uni 3572
This theorem is referenced by: (None)
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