Theorem nlimsucg 7793

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 6384	. . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 7765	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 234	. . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 6385	. . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5		ordunisuc 7783	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
6	5	eqeq2d 2747	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
7		ordirr 6341	. . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
8		eleq2 2825	. . . . . . 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 6406	. . . . . 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 1542   ∈ wcel 2114  ∪ cuni 4850  Ord word 6322  Lim wlim 6324  suc csuc 6325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329

This theorem is referenced by:  tz7.44-2  8346  rankxpsuc  9806  cutbdaybnd2lim  27789  dfrdg2  35975  dfrdg4  36133  onov0suclim  43702  dflim5  43757