Theorem nlimsuc 43900

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 6397	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		eloni 6324	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordirr 6332	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
5		eleq2 2830	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
6	5	notbid 320	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
7	4, 6	syl5ibrcom 249	. . . . . . 7 ⊢ (𝐴 ∈ On → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
8	1, 7	mt2d 136	. . . . . 6 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = 𝐴)
9	8	neqned 2943	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ 𝐴)
10		onunisuc 6426	. . . . 5 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
11	9, 10	neeqtrrd 3010	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ ∪ suc 𝐴)
12	11	neneqd 2941	. . 3 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = ∪ suc 𝐴)
13	12	intn3an3d 1490	. 2 ⊢ (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
14		dflim2 6372	. 2 ⊢ (Lim suc 𝐴 ↔ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
15	13, 14	sylnibr 331	1 ⊢ (𝐴 ∈ On → ¬ Lim suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∅c0 4264  ∪ cuni 4841  Ord word 6313  Oncon0 6314  Lim wlim 6315  suc csuc 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-tr 5183  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320

This theorem is referenced by:  nlim1NEW  43901  nlim2NEW  43902  nlim3  43903  nlim4  43904  dfsucon  43982