Theorem nlim2NEW 43447

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 8523	. 2 ⊢ 1o ∈ On
2		nlimsuc 43445	. . 3 ⊢ (1o ∈ On → ¬ Lim suc 1o)
3		df-2o 8512	. . . 4 ⊢ 2o = suc 1o
4		limeq 6401	. . . 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 1538   ∈ wcel 2107  Oncon0 6389  Lim wlim 6390  suc csuc 6391  1_oc1o 8504  2_oc2o 8505

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 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-pss 3984  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5590  df-po 5598  df-so 5599  df-fr 5642  df-we 5644  df-ord 6392  df-on 6393  df-lim 6394  df-suc 6395  df-1o 8511  df-2o 8512

This theorem is referenced by: (None)