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Theorem nlimsucg 7828
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
nlimsucg (𝐴𝑉 → ¬ Lim suc 𝐴)

Proof of Theorem nlimsucg
StepHypRef Expression
1 limord 6418 . . . 4 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 7798 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 233 . . 3 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 6419 . . 3 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
5 ordunisuc 7817 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
65eqeq2d 2737 . . . 4 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
7 ordirr 6376 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
8 eleq2 2816 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
98notbid 318 . . . . . 6 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
107, 9syl5ibrcom 246 . . . . 5 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11 sucidg 6439 . . . . . 6 (𝐴𝑉𝐴 ∈ suc 𝐴)
1211con3i 154 . . . . 5 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1310, 12syl6 35 . . . 4 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
146, 13sylbid 239 . . 3 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
153, 4, 14sylc 65 . 2 (Lim suc 𝐴 → ¬ 𝐴𝑉)
1615con2i 139 1 (𝐴𝑉 → ¬ Lim suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098   cuni 4902  Ord word 6357  Lim wlim 6359  suc csuc 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364
This theorem is referenced by:  tz7.44-2  8408  rankxpsuc  9879  scutbdaybnd2lim  27705  dfrdg2  35300  dfrdg4  35456  onov0suclim  42597  dflim5  42652
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