Theorem nlim2NEW 43407

Description: 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)

Assertion

Ref	Expression
nlim2NEW	⊢ ¬ Lim 2o

Proof of Theorem nlim2NEW

Step	Hyp	Ref	Expression
1		1on 8536	. 2 ⊢ 1o ∈ On
2		nlimsuc 43405	. . 3 ⊢ (1o ∈ On → ¬ Lim suc 1o)
3		df-2o 8525	. . . 4 ⊢ 2o = suc 1o
4		limeq 6409	. . . 4 ⊢ (2o = suc 1o → (Lim 2o ↔ Lim suc 1o))
5	3, 4	ax-mp 5	. . 3 ⊢ (Lim 2o ↔ Lim suc 1o)
6	2, 5	sylnibr 329	. 2 ⊢ (1o ∈ On → ¬ Lim 2o)
7	1, 6	ax-mp 5	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2108  Oncon0 6397  Lim wlim 6398  suc csuc 6399  1_oc1o 8517  2_oc2o 8518

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 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

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 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-1o 8524  df-2o 8525

This theorem is referenced by: (None)