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Theorem nlim0 6444
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 4343 . . 3 ¬ ∅ ∈ ∅
2 simp2 1136 . . 3 ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅) → ∅ ∈ ∅)
31, 2mto 197 . 2 ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅)
4 dflim2 6442 . 2 (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅))
53, 4mtbir 323 1 ¬ Lim ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1086   = wceq 1536  wcel 2105  c0 4338   cuni 4911  Ord word 6384  Lim wlim 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-lim 6390
This theorem is referenced by:  0ellim  6448  tz7.44lem1  8443  tz7.44-3  8446  1ellim  8534  2ellim  8535  cflim2  10300  rankcf  10814  dfrdg4  35932  limsucncmpi  36427  onov0suclim  43263
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