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Theorem nlimsuc 43431
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 6467 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6396 . . . . . . . . 9 (𝐴 ∈ On → Ord 𝐴)
3 ordirr 6404 . . . . . . . . 9 (Ord 𝐴 → ¬ 𝐴𝐴)
42, 3syl 17 . . . . . . . 8 (𝐴 ∈ On → ¬ 𝐴𝐴)
5 eleq2 2828 . . . . . . . . 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 2945 . . . . 5 (𝐴 ∈ On → suc 𝐴𝐴)
10 onunisuc 6496 . . . . 5 (𝐴 ∈ On → suc 𝐴 = 𝐴)
119, 10neeqtrrd 3013 . . . 4 (𝐴 ∈ On → suc 𝐴 suc 𝐴)
1211neneqd 2943 . . 3 (𝐴 ∈ On → ¬ suc 𝐴 = suc 𝐴)
1312intn3an3d 1480 . 2 (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = suc 𝐴))
14 dflim2 6443 . 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 1537  wcel 2106  c0 4339   cuni 4912  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388
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
This theorem is referenced by:  nlim1NEW  43432  nlim2NEW  43433  nlim3  43434  nlim4  43435  dfsucon  43513
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