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Theorem nlim0 4212
Description: The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
nlim0  |-  -.  Lim  (/)

Proof of Theorem nlim0
StepHypRef Expression
1 noel 3288 . . 3  |-  -.  (/)  e.  (/)
2 simp2 944 . . 3  |-  ( ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )  ->  (/) 
e.  (/) )
31, 2mto 623 . 2  |-  -.  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) )
4 dflim2 4188 . 2  |-  ( Lim  (/) 
<->  ( Ord  (/)  /\  (/)  e.  (/)  /\  (/)  =  U. (/) ) )
53, 4mtbir 631 1  |-  -.  Lim  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ w3a 924    = wceq 1289    e. wcel 1438   (/)c0 3284   U.cuni 3648   Ord word 4180   Lim wlim 4182
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-ilim 4187
This theorem is referenced by: (None)
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