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Theorem nlim1NEW 41788
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 6376 . 2 ∅ ∈ On
2 nlimsuc 41787 . . 3 (∅ ∈ On → ¬ Lim suc ∅)
3 df-1o 8417 . . . 4 1o = suc ∅
4 limeq 6334 . . . 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 1542  wcel 2107  c0 4287  Oncon0 6322  Lim wlim 6323  suc csuc 6324  1oc1o 8410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-1o 8417
This theorem is referenced by: (None)
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