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Theorem nlim0 4379
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 3418 . . 3 ¬ ∅ ∈ ∅
2 simp2 993 . . 3 ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅) → ∅ ∈ ∅)
31, 2mto 657 . 2 ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅)
4 dflim2 4355 . 2 (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∅))
53, 4mtbir 666 1 ¬ Lim ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  w3a 973   = wceq 1348  wcel 2141  c0 3414   cuni 3796  Ord word 4347  Lim wlim 4349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-ilim 4354
This theorem is referenced by: (None)
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