Theorem ordsuci 7762

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 7764. (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 6337	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		suctr 6411	. . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (Ord 𝐴 → Tr suc 𝐴)
4		df-suc 6329	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		ordsson 7737	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
6		elon2 6334	. . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))
7		snssi 4729	. . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
8	6, 7	sylbir 235	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On)
9		snprc 4661	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 216	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11		0ss 4340	. . . . . . 7 ⊢ ∅ ⊆ On
12	10, 11	eqsstrdi 3966	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On)
13	12	adantl 481	. . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
14	8, 13	pm2.61dan 813	. . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On)
15	5, 14	unssd 4132	. . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
16	4, 15	eqsstrid 3960	. 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On)
17		ordon 7731	. . 3 ⊢ Ord On
18	17	a1i 11	. 2 ⊢ (Ord 𝐴 → Ord On)
19		trssord 6340	. 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
20	3, 16, 18, 19	syl3anc 1374	1 ⊢ (Ord 𝐴 → Ord suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  {csn 4567  Tr wtr 5192  Ord word 6322  Oncon0 6323  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

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-suc 6329

This theorem is referenced by:  sucexeloni  7763  ordsuc  7765  ord3  8420  ordeldifsucon  43687  ordeldif1o  43688  ordnexbtwnsuc  43695  ordsssucb  43763  onsucunifi  43798