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Theorem nlim4 41707
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 8430 . 2 3o ∈ On
2 nlimsuc 41703 . . 3 (3o ∈ On → ¬ Lim suc 3o)
3 df-4o 8415 . . . 4 4o = suc 3o
4 limeq 6329 . . . 4 (4o = suc 3o → (Lim 4o ↔ Lim suc 3o))
53, 4ax-mp 5 . . 3 (Lim 4o ↔ Lim suc 3o)
62, 5sylnibr 328 . 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 1541  wcel 2106  Oncon0 6317  Lim wlim 6318  suc csuc 6319  3oc3o 8407  4oc4o 8408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-tr 5223  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-1o 8412  df-2o 8413  df-3o 8414  df-4o 8415
This theorem is referenced by: (None)
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