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Theorem uni0 3657
Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
Assertion
Ref Expression
uni0  |-  U. (/)  =  (/)

Proof of Theorem uni0
StepHypRef Expression
1 0ss 3306 . 2  |-  (/)  C_  { (/) }
2 uni0b 3655 . 2  |-  ( U. (/)  =  (/)  <->  (/)  C_  { (/) } )
31, 2mpbir 144 1  |-  U. (/)  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1287    C_ wss 2986   (/)c0 3272   {csn 3425   U.cuni 3630
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-nul 3273  df-sn 3431  df-uni 3631
This theorem is referenced by:  iununir  3788  unixp0im  4924  iotanul  4952  1st0  5853  2nd0  5854  brtpos0  5952  tpostpos  5964  nnsucuniel  6191  sup00  6619  nnpredcl  11246
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