Theorem nlimsuc 42765

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 6439	. . . . . . 7 ⊢ (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2		eloni 6368	. . . . . . . . 9 ⊢ (𝐴 ∈ On → Ord 𝐴)
3		ordirr 6376	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
4	2, 3	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
5		eleq2 2816	. . . . . . . . 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 246	. . . . . . 7 ⊢ (𝐴 ∈ On → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
8	1, 7	mt2d 136	. . . . . 6 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = 𝐴)
9	8	neqned 2941	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ 𝐴)
10		onunisuc 6468	. . . . 5 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
11	9, 10	neeqtrrd 3009	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ≠ ∪ suc 𝐴)
12	11	neneqd 2939	. . 3 ⊢ (𝐴 ∈ On → ¬ suc 𝐴 = ∪ suc 𝐴)
13	12	intn3an3d 1477	. 2 ⊢ (𝐴 ∈ On → ¬ (Ord suc 𝐴 ∧ ∅ ∈ suc 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
14		dflim2 6415	. 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 1084   = wceq 1533   ∈ wcel 2098  ∅c0 4317  ∪ cuni 4902  Ord word 6357  Oncon0 6358  Lim wlim 6359  suc csuc 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364

This theorem is referenced by:  nlim1NEW  42766  nlim2NEW  42767  nlim3  42768  nlim4  42769  dfsucon  42847