Theorem nlim2 8471

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

Assertion

Ref	Expression
nlim2	⊢ ¬ Lim 2o

Proof of Theorem nlim2

Step	Hyp	Ref	Expression
1		1oelpr 8460	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
2		df2o3 8457	. . . . . . . 8 ⊢ 2o = {∅, 1o}
3	1, 2	eleqtrri 2862	. . . . . . 7 ⊢ 1o ∈ 2o
4		1on 8462	. . . . . . . . 9 ⊢ 1o ∈ On
5	4	onirri 6475	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
6		eleq2 2852	. . . . . . . 8 ⊢ (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o))
7	5, 6	mtbiri 330	. . . . . . 7 ⊢ (2o = 1o → ¬ 1o ∈ 2o)
8	3, 7	mt2 203	. . . . . 6 ⊢ ¬ 2o = 1o
9	8	neir 2961	. . . . 5 ⊢ 2o ≠ 1o
10	2	unieqi 4884	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
11		0ex 5270	. . . . . . 7 ⊢ ∅ ∈ V
12		1oex 8459	. . . . . . 7 ⊢ 1o ∈ V
13	11, 12	unipr 4889	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4353	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	10, 13, 14	3eqtri 2790	. . . . 5 ⊢ ∪ 2o = 1o
16	9, 15	neeqtrri 3031	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2960	. . 3 ⊢ ¬ 2o = ∪ 2o
18		simp3 1156	. . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o)
19	17, 18	mto 200	. 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)
20		df-lim 6365	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 326	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∪ cun 3903  ∅c0 4286  {cpr 4591  ∪ cuni 4872  Ord word 6359  Lim wlim 6361  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 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-1o 8449  df-2o 8450

This theorem is referenced by:  2ellim  8480