Theorem nlim2 8527

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 8515	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4768	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8513	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2838	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8517	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6499	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2828	. . . . . . . 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 2941	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4924	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5313	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4929	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4402	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2767	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 3012	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2940	. . 3 ⊢ ¬ 2o = ∪ 2o
18		simp3 1137	. . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o)
19	17, 18	mto 197	. 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)
20		df-lim 6391	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 323	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∪ cun 3961  ∅c0 4339  {cpr 4633  ∪ cuni 4912  Ord word 6385  Lim wlim 6387  1_oc1o 8498  2_oc2o 8499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-1o 8505  df-2o 8506

This theorem is referenced by:  2ellim  8536