Theorem nlim2 8405

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 8395	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4713	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8393	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2830	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8397	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6420	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2820	. . . . . . . 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 2931	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4868	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5243	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4873	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4343	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2758	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 3001	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2930	. . 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 6311	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 323	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   ∪ cun 3895  ∅c0 4280  {cpr 4575  ∪ cuni 4856  Ord word 6305  Lim wlim 6307  1_oc1o 8378  2_oc2o 8379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-1o 8385  df-2o 8386

This theorem is referenced by:  2ellim  8414