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Theorem uni0 3654
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 ∅ = ∅

Proof of Theorem uni0
StepHypRef Expression
1 0ss 3303 . 2 ∅ ⊆ {∅}
2 uni0b 3652 . 2 ( ∅ = ∅ ↔ ∅ ⊆ {∅})
31, 2mpbir 144 1 ∅ = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wss 2984  c0 3269  {csn 3422   cuni 3627
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-uni 3628
This theorem is referenced by:  iununir  3785  unixp0im  4919  iotanul  4947  1st0  5848  2nd0  5849  brtpos0  5947  tpostpos  5959  nnsucuniel  6186  sup00  6603
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