Theorem nlim2 8408

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 8398	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4715	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8396	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2827	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8400	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6421	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2817	. . . . . . . 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 2928	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4870	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5246	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4875	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4347	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2756	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 2998	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2927	. . 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 6312	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 323	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∪ cun 3901  ∅c0 4284  {cpr 4579  ∪ cuni 4858  Ord word 6306  Lim wlim 6308  1_oc1o 8381  2_oc2o 8382

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-1o 8388  df-2o 8389

This theorem is referenced by:  2ellim  8417