Theorem nlim2NEW 44054

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 8462	. 2 ⊢ 1o ∈ On
2		nlimsuc 44052	. . 3 ⊢ (1o ∈ On → ¬ Lim suc 1o)
3		df-2o 8450	. . . 4 ⊢ 2o = suc 1o
4		limeq 6369	. . . 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 332	. 2 ⊢ (1o ∈ On → ¬ Lim 2o)
7	1, 6	ax-mp 5	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1567   ∈ wcel 2149  Oncon0 6357  Lim wlim 6358  suc csuc 6359  1_oc1o 8442  2_oc2o 8443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-1o 8449  df-2o 8450

This theorem is referenced by: (None)