Theorem nlimsuc 43403

Description: A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)

Assertion

Ref	Expression
nlimsuc	⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)

Proof of Theorem nlimsuc

Step	Hyp	Ref	Expression
1		sucidg 6476	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		eloni 6405	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordirr 6413	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
5		eleq2 2833	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
6	5	notbid 318	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
7	4, 6	syl5ibrcom 247	. . . . . . 7 ⊢ (𝐴 ∈ On → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
8	1, 7	mt2d 136	. . . . . 6 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = 𝐴)
9	8	neqned 2953	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ 𝐴)
10		onunisuc 6505	. . . . 5 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
11	9, 10	neeqtrrd 3021	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ ∪ suc 𝐴)
12	11	neneqd 2951	. . 3 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = ∪ suc 𝐴)
13	12	intn3an3d 1481	. 2 ⊢ (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
14		dflim2 6452	. 2 ⊢ (Lim suc 𝐴 ↔ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
15	13, 14	sylnibr 329	1 ⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∅c0 4352  ∪ cuni 4931  Ord word 6394  Oncon0 6395  Lim wlim 6396  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401

This theorem is referenced by:  nlim1NEW  43404  nlim2NEW  43405  nlim3  43406  nlim4  43407  dfsucon  43485