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Theorem nlimsucg 7863
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 6444 . . . 4 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 7833 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 234 . . 3 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 6445 . . 3 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
5 ordunisuc 7852 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
65eqeq2d 2748 . . . 4 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
7 ordirr 6402 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
8 eleq2 2830 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
98notbid 318 . . . . . 6 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
107, 9syl5ibrcom 247 . . . . 5 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11 sucidg 6465 . . . . . 6 (𝐴𝑉𝐴 ∈ suc 𝐴)
1211con3i 154 . . . . 5 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1310, 12syl6 35 . . . 4 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
146, 13sylbid 240 . . 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 1540  wcel 2108   cuni 4907  Ord word 6383  Lim wlim 6385  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390
This theorem is referenced by:  tz7.44-2  8447  rankxpsuc  9922  scutbdaybnd2lim  27862  dfrdg2  35796  dfrdg4  35952  onov0suclim  43287  dflim5  43342
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