Theorem ordsuci 7792

Description: The successor of an ordinal class is an ordinal class. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) Extract and adapt from a subproof of onsuc 7795. (Revised by BTernaryTau, 6-Jan-2025.) (Proof shortened by BJ, 11-Jan-2025.)

Assertion

Ref	Expression
ordsuci	⊢ (Ord 𝐴 → Ord suc 𝐴)

Proof of Theorem ordsuci

Step	Hyp	Ref	Expression
1		ordtr 6371	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		suctr 6443	. . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (Ord 𝐴 → Tr suc 𝐴)
4		df-suc 6363	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		ordsson 7766	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
6		elon2 6368	. . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))
7		snssi 4806	. . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
8	6, 7	sylbir 234	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On)
9		snprc 4716	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11		0ss 4391	. . . . . . 7 ⊢ ∅ ⊆ On
12	10, 11	eqsstrdi 4031	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On)
13	12	adantl 481	. . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
14	8, 13	pm2.61dan 810	. . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On)
15	5, 14	unssd 4181	. . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
16	4, 15	eqsstrid 4025	. 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On)
17		ordon 7760	. . 3 ⊢ Ord On
18	17	a1i 11	. 2 ⊢ (Ord 𝐴 → Ord On)
19		trssord 6374	. 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
20	3, 16, 18, 19	syl3anc 1368	1 ⊢ (Ord 𝐴 → Ord suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∪ cun 3941   ⊆ wss 3943  ∅c0 4317  {csn 4623  Tr wtr 5258  Ord word 6356  Oncon0 6357  suc csuc 6359

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 6360  df-on 6361  df-suc 6363

This theorem is referenced by:  sucexeloni  7793  ordsuc  7797  ord3  8481  ordeldifsucon  42567  ordeldif1o  42568  ordnexbtwnsuc  42575  ordsssucb  42643  onsucunifi  42678