Theorem nlim2 8528

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 8516	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4763	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8514	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2840	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8518	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6497	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2830	. . . . . . . 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 2943	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4919	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5307	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4924	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4396	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2769	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 3014	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2942	. . 3 ⊢ ¬ 2o = ∪ 2o
18		simp3 1139	. . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o)
19	17, 18	mto 197	. 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)
20		df-lim 6389	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 323	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940   ∪ cun 3949  ∅c0 4333  {cpr 4628  ∪ cuni 4907  Ord word 6383  Lim wlim 6385  1_oc1o 8499  2_oc2o 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-1o 8506  df-2o 8507

This theorem is referenced by:  2ellim  8537