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Theorem nlim0 4097
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 3222 . . 3 ¬ ∅
2 simp2 904 . . 3 ((Ord ∅ ∅ = ∅) → ∅ ∅)
31, 2mto 587 . 2 ¬ (Ord ∅ ∅ = ∅)
4 dflim2 4073 . 2 (Lim ∅ ↔ (Ord ∅ ∅ = ∅))
53, 4mtbir 595 1 ¬ Lim ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   w3a 884   = wceq 1242   wcel 1390  c0 3218   cuni 3571  Ord word 4065  Lim wlim 4067
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-bndl 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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-ilim 4072
This theorem is referenced by: (None)
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