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Theorem nlim1NEW 42751
Description: 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
Assertion
Ref Expression
nlim1NEW ¬ Lim 1o

Proof of Theorem nlim1NEW
StepHypRef Expression
1 0elon 6411 . 2 ∅ ∈ On
2 nlimsuc 42750 . . 3 (∅ ∈ On → ¬ Lim suc ∅)
3 df-1o 8464 . . . 4 1o = suc ∅
4 limeq 6369 . . . 4 (1o = suc ∅ → (Lim 1o ↔ Lim suc ∅))
53, 4ax-mp 5 . . 3 (Lim 1o ↔ Lim suc ∅)
62, 5sylnibr 329 . 2 (∅ ∈ On → ¬ Lim 1o)
71, 6ax-mp 5 1 ¬ Lim 1o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  c0 4317  Oncon0 6357  Lim wlim 6358  suc csuc 6359  1oc1o 8457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-1o 8464
This theorem is referenced by: (None)
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