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Theorem nlim4 43348
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 8536 . 2 3o ∈ On
2 nlimsuc 43344 . . 3 (3o ∈ On → ¬ Lim suc 3o)
3 df-4o 8521 . . . 4 4o = suc 3o
4 limeq 6406 . . . 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 206   = wceq 1537  wcel 2103  Oncon0 6394  Lim wlim 6395  suc csuc 6396  3oc3o 8513  4oc4o 8514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-tr 5287  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-1o 8518  df-2o 8519  df-3o 8520  df-4o 8521
This theorem is referenced by: (None)
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