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Theorem nlimsucg 7276
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 6000 . . . 4 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 7248 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 226 . . 3 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 6001 . . 3 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
5 ordunisuc 7266 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
65eqeq2d 2809 . . . 4 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
7 ordirr 5959 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
8 eleq2 2867 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
98notbid 310 . . . . . 6 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
107, 9syl5ibrcom 239 . . . . 5 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11 sucidg 6019 . . . . . 6 (𝐴𝑉𝐴 ∈ suc 𝐴)
1211con3i 152 . . . . 5 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1310, 12syl6 35 . . . 4 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
146, 13sylbid 232 . . 3 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
153, 4, 14sylc 65 . 2 (Lim suc 𝐴 → ¬ 𝐴𝑉)
1615con2i 137 1 (𝐴𝑉 → ¬ Lim suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1653  wcel 2157   cuni 4628  Ord word 5940  Lim wlim 5942  suc csuc 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947
This theorem is referenced by:  tz7.44-2  7742  rankxpsuc  8995  dfrdg2  32213  dfrdg4  32571
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