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Theorem nlim0 6443
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 4338 . . 3 ¬ ∅ ∈ ∅
2 simp2 1138 . . 3 ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅) → ∅ ∈ ∅)
31, 2mto 197 . 2 ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅)
4 dflim2 6441 . 2 (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅))
53, 4mtbir 323 1 ¬ Lim ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1087   = wceq 1540  wcel 2108  c0 4333   cuni 4907  Ord word 6383  Lim wlim 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-lim 6389
This theorem is referenced by:  0ellim  6447  tz7.44lem1  8445  tz7.44-3  8448  1ellim  8536  2ellim  8537  cflim2  10303  rankcf  10817  dfrdg4  35952  limsucncmpi  36446  onov0suclim  43287
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