Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nlim0 Structured version   Visualization version   GIF version

Theorem nlim0 6224
 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 4250 . . 3 ¬ ∅ ∈ ∅
2 simp2 1134 . . 3 ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅) → ∅ ∈ ∅)
31, 2mto 200 . 2 ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅)
4 dflim2 6222 . 2 (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅))
53, 4mtbir 326 1 ¬ Lim ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∅c0 4246  ∪ cuni 4804  Ord word 6165  Lim wlim 6167 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-tr 5141  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6169  df-lim 6171 This theorem is referenced by:  0ellim  6228  tz7.44lem1  8042  tz7.44-3  8045  cflim2  9692  rankcf  10206  dfrdg4  33672  limsucncmpi  34053
 Copyright terms: Public domain W3C validator