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Theorem nlimsuc 43459
Description: A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
Assertion
Ref Expression
nlimsuc (𝐴 ∈ On → ¬ Lim suc 𝐴)

Proof of Theorem nlimsuc
StepHypRef Expression
1 sucidg 6464 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6393 . . . . . . . . 9 (𝐴 ∈ On → Ord 𝐴)
3 ordirr 6401 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3syl 17 . . . . . . . 8 (𝐴 ∈ On → ¬ 𝐴𝐴)
5 eleq2 2829 . . . . . . . . 9 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
65notbid 318 . . . . . . . 8 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
74, 6syl5ibrcom 247 . . . . . . 7 (𝐴 ∈ On → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
81, 7mt2d 136 . . . . . 6 (𝐴 ∈ On → ¬ suc 𝐴 = 𝐴)
98neqned 2946 . . . . 5 (𝐴 ∈ On → suc 𝐴𝐴)
10 onunisuc 6493 . . . . 5 (𝐴 ∈ On → suc 𝐴 = 𝐴)
119, 10neeqtrrd 3014 . . . 4 (𝐴 ∈ On → suc 𝐴 suc 𝐴)
1211neneqd 2944 . . 3 (𝐴 ∈ On → ¬ suc 𝐴 = suc 𝐴)
1312intn3an3d 1482 . 2 (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = suc 𝐴))
14 dflim2 6440 . 2 (Lim suc 𝐴 ↔ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = suc 𝐴))
1513, 14sylnibr 329 1 (𝐴 ∈ On → ¬ Lim suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  w3a 1086   = wceq 1539  wcel 2107  c0 4332   cuni 4906  Ord word 6382  Oncon0 6383  Lim wlim 6384  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389
This theorem is referenced by:  nlim1NEW  43460  nlim2NEW  43461  nlim3  43462  nlim4  43463  dfsucon  43541
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