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Theorem nlim0 5944
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 4062 . . 3 ¬ ∅ ∈ ∅
2 simp2 1132 . . 3 ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅) → ∅ ∈ ∅)
31, 2mto 188 . 2 ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅)
4 dflim2 5942 . 2 (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅))
53, 4mtbir 312 1 ¬ Lim ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1072   = wceq 1632  wcel 2139  c0 4058   cuni 4588  Ord word 5883  Lim wlim 5885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887  df-lim 5889
This theorem is referenced by:  0ellim  5948  tz7.44lem1  7671  tz7.44-3  7674  cflim2  9297  rankcf  9811  dfrdg4  32385  limsucncmpi  32771
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