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Theorem nlim4 42869
Description: 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
Assertion
Ref Expression
nlim4 ¬ Lim 4o

Proof of Theorem nlim4
StepHypRef Expression
1 3on 8498 . 2 3o ∈ On
2 nlimsuc 42865 . . 3 (3o ∈ On → ¬ Lim suc 3o)
3 df-4o 8483 . . . 4 4o = suc 3o
4 limeq 6375 . . . 4 (4o = suc 3o → (Lim 4o ↔ Lim suc 3o))
53, 4ax-mp 5 . . 3 (Lim 4o ↔ Lim suc 3o)
62, 5sylnibr 329 . 2 (3o ∈ On → ¬ Lim 4o)
71, 6ax-mp 5 1 ¬ Lim 4o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1534  wcel 2099  Oncon0 6363  Lim wlim 6364  suc csuc 6365  3oc3o 8475  4oc4o 8476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-1o 8480  df-2o 8481  df-3o 8482  df-4o 8483
This theorem is referenced by: (None)
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