Theorem nlimsucg 7853

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 6424	. . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 7825	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 237	. . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 6425	. . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5		ordunisuc 7843	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
6	5	eqeq2d 2772	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
7		ordirr 6380	. . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
8		eleq2 2850	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
9	8	notbid 321	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
10	7, 9	syl5ibrcom 250	. . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11		sucidg 6446	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	con3i 155	. . . . 5 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
13	10, 12	syl6 36	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
14	6, 13	sylbid 243	. . 3 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉))
15	3, 4, 14	sylc 66	. 2 ⊢ (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
16	15	con2i 140	1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  ∪ cuni 4867  Ord word 6361  Lim wlim 6363  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368

This theorem is used by:  tz7.44-2  8415  rankxpsuc  9899  cutbdaybnd2lim  28183  dfrdg2  36557  dfrdg4  36715  onov0suclim  44275  dflim5  44330