Theorem nlim2 8546

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		1oex 8532	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4788	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8530	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2843	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8534	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6508	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2833	. . . . . . . 8 ⊢ (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o))
8	6, 7	mtbiri 327	. . . . . . 7 ⊢ (2o = 1o → ¬ 1o ∈ 2o)
9	4, 8	mt2 200	. . . . . 6 ⊢ ¬ 2o = 1o
10	9	neir 2949	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4943	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5325	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4948	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4419	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2772	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 3020	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2948	. . 3 ⊢ ¬ 2o = ∪ 2o
18		simp3 1138	. . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o)
19	17, 18	mto 197	. 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)
20		df-lim 6400	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 323	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∪ cun 3974  ∅c0 4352  {cpr 4650  ∪ cuni 4931  Ord word 6394  Lim wlim 6396  1_oc1o 8515  2_oc2o 8516

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 6398  df-on 6399  df-lim 6400  df-suc 6401  df-1o 8522  df-2o 8523

This theorem is referenced by:  2ellim  8555