Theorem nlim2 8459

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 8447	. . . . . . . . 9 ⊢ 1o ∈ V
2	1	prid2 4722	. . . . . . . 8 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8445	. . . . . . . 8 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2861	. . . . . . 7 ⊢ 1o ∈ 2o
5		1on 8450	. . . . . . . . 9 ⊢ 1o ∈ On
6	5	onirri 6460	. . . . . . . 8 ⊢ ¬ 1o ∈ 1o
7		eleq2 2851	. . . . . . . 8 ⊢ (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o))
8	6, 7	mtbiri 329	. . . . . . 7 ⊢ (2o = 1o → ¬ 1o ∈ 2o)
9	4, 8	mt2 202	. . . . . 6 ⊢ ¬ 2o = 1o
10	9	neir 2960	. . . . 5 ⊢ 2o ≠ 1o
11	3	unieqi 4877	. . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o}
12		0ex 5257	. . . . . . 7 ⊢ ∅ ∈ V
13	12, 1	unipr 4882	. . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o)
14		0un 4350	. . . . . 6 ⊢ (∅ ∪ 1o) = 1o
15	11, 13, 14	3eqtri 2789	. . . . 5 ⊢ ∪ 2o = 1o
16	10, 15	neeqtrri 3030	. . . 4 ⊢ 2o ≠ ∪ 2o
17	16	neii 2959	. . 3 ⊢ ¬ 2o = ∪ 2o
18		simp3 1151	. . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o)
19	17, 18	mto 199	. 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)
20		df-lim 6351	. 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o))
21	19, 20	mtbir 325	1 ⊢ ¬ Lim 2o

Syntax hints:  ¬ wn 3   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   ∪ cun 3902  ∅c0 4285  {cpr 4584  ∪ cuni 4865  Ord word 6345  Lim wlim 6347  1_oc1o 8430  2_oc2o 8431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-1o 8437  df-2o 8438

This theorem is referenced by:  2ellim  8468