Theorem nlim2 8492

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 8478	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4767	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8476	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2832	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8480	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6477	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2822	. . . . . . . 8 ⊢ (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o))
8	6, 7	mtbiri 326	. . . . . . 7 ⊢ (2o = 1o → ¬ 1o ∈ 2o)
9	4, 8	mt2 199	. . . . . 6 ⊢ ¬ 2o = 1o
10	9	neir 2943	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4921	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5307	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4926	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4392	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2764	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 3014	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2942	. . 3 ⊢ ¬ 2o = ∪ 2o
18		simp3 1138	. . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o)
19	17, 18	mto 196	. 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)
20		df-lim 6369	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 322	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   ∪ cun 3946  ∅c0 4322  {cpr 4630  ∪ cuni 4908  Ord word 6363  Lim wlim 6365  1_oc1o 8461  2_oc2o 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-1o 8468  df-2o 8469

This theorem is referenced by:  2ellim  8501