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Theorem nlim1NEW 42903
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 6428 . 2 ∅ ∈ On
2 nlimsuc 42902 . . 3 (∅ ∈ On → ¬ Lim suc ∅)
3 df-1o 8493 . . . 4 1o = suc ∅
4 limeq 6386 . . . 4 (1o = suc ∅ → (Lim 1o ↔ Lim suc ∅))
53, 4ax-mp 5 . . 3 (Lim 1o ↔ Lim suc ∅)
62, 5sylnibr 328 . 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 4326  Oncon0 6374  Lim wlim 6375  suc csuc 6376  1oc1o 8486
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-1o 8493
This theorem is referenced by: (None)
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