Theorem nlimsucg 7782

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	⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)

Proof of Theorem nlimsucg

Step	Hyp	Ref	Expression
1		limord 6376	. . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 7754	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 234	. . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 6377	. . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5		ordunisuc 7772	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
6	5	eqeq2d 2745	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
7		ordirr 6333	. . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
8		eleq2 2823	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
9	8	notbid 318	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
10	7, 9	syl5ibrcom 247	. . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11		sucidg 6398	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
13	10, 12	syl6 35	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
14	6, 13	sylbid 240	. . 3 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉))
15	3, 4, 14	sylc 65	. 2 ⊢ (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
16	15	con2i 139	1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  ∪ cuni 4861  Ord word 6314  Lim wlim 6316  suc csuc 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321

This theorem is referenced by:  tz7.44-2  8336  rankxpsuc  9792  scutbdaybnd2lim  27785  dfrdg2  35936  dfrdg4  36094  onov0suclim  43458  dflim5  43513