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Theorem nlimsuc 41787
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 6403 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6332 . . . . . . . . 9 (𝐴 ∈ On → Ord 𝐴)
3 ordirr 6340 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3syl 17 . . . . . . . 8 (𝐴 ∈ On → ¬ 𝐴𝐴)
5 eleq2 2827 . . . . . . . . 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 2951 . . . . 5 (𝐴 ∈ On → suc 𝐴𝐴)
10 onunisuc 6432 . . . . 5 (𝐴 ∈ On → suc 𝐴 = 𝐴)
119, 10neeqtrrd 3019 . . . 4 (𝐴 ∈ On → suc 𝐴 suc 𝐴)
1211neneqd 2949 . . 3 (𝐴 ∈ On → ¬ suc 𝐴 = suc 𝐴)
1312intn3an3d 1482 . 2 (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = suc 𝐴))
14 dflim2 6379 . 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 1088   = wceq 1542  wcel 2107  c0 4287   cuni 4870  Ord word 6321  Oncon0 6322  Lim wlim 6323  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328
This theorem is referenced by:  nlim1NEW  41788  nlim2NEW  41789  nlim3  41790  nlim4  41791  dfsucon  41869
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