Theorem nlimsucg 7831

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 6425	. . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 7801	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 233	. . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 6426	. . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5		ordunisuc 7820	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
6	5	eqeq2d 2744	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
7		ordirr 6383	. . . . . 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 246	. . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11		sucidg 6446	. . . . . 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 239	. . 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 1542   ∈ wcel 2107  ∪ cuni 4909  Ord word 6364  Lim wlim 6366  suc csuc 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371

This theorem is referenced by:  tz7.44-2  8407  rankxpsuc  9877  scutbdaybnd2lim  27318  dfrdg2  34767  dfrdg4  34923  onov0suclim  42024  dflim5  42079