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Theorem nlim2 8528
Description: 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.)
Assertion
Ref Expression
nlim2 ¬ Lim 2o

Proof of Theorem nlim2
StepHypRef Expression
1 1oex 8516 . . . . . . . . 9 1o ∈ V
21prid2 4763 . . . . . . . 8 1o ∈ {∅, 1o}
3 df2o3 8514 . . . . . . . 8 2o = {∅, 1o}
42, 3eleqtrri 2840 . . . . . . 7 1o ∈ 2o
5 1on 8518 . . . . . . . . 9 1o ∈ On
65onirri 6497 . . . . . . . 8 ¬ 1o ∈ 1o
7 eleq2 2830 . . . . . . . 8 (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o))
86, 7mtbiri 327 . . . . . . 7 (2o = 1o → ¬ 1o ∈ 2o)
94, 8mt2 200 . . . . . 6 ¬ 2o = 1o
109neir 2943 . . . . 5 2o ≠ 1o
113unieqi 4919 . . . . . 6 2o = {∅, 1o}
12 0ex 5307 . . . . . . 7 ∅ ∈ V
1312, 1unipr 4924 . . . . . 6 {∅, 1o} = (∅ ∪ 1o)
14 0un 4396 . . . . . 6 (∅ ∪ 1o) = 1o
1511, 13, 143eqtri 2769 . . . . 5 2o = 1o
1610, 15neeqtrri 3014 . . . 4 2o 2o
1716neii 2942 . . 3 ¬ 2o = 2o
18 simp3 1139 . . 3 ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = 2o) → 2o = 2o)
1917, 18mto 197 . 2 ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = 2o)
20 df-lim 6389 . 2 (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = 2o))
2119, 20mtbir 323 1 ¬ Lim 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cun 3949  c0 4333  {cpr 4628   cuni 4907  Ord word 6383  Lim wlim 6385  1oc1o 8499  2oc2o 8500
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-1o 8506  df-2o 8507
This theorem is referenced by:  2ellim  8537
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