Theorem ordsuci 7795

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 7797. (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 6363	. . 3 ⊢ (Ord 𝐴 → Tr 𝐴)
2		suctr 6438	. . 3 ⊢ (Tr 𝐴 → Tr suc 𝐴)
3	1, 2	syl 18	. 2 ⊢ (Ord 𝐴 → Tr suc 𝐴)
4		df-suc 6355	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5		ordsson 7770	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
6		elon2 6360	. . . . . 6 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))
7		snssi 4747	. . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
8	6, 7	sylbir 238	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) → {𝐴} ⊆ On)
9		snprc 4679	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 219	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11		0ss 4357	. . . . . . 7 ⊢ ∅ ⊆ On
12	10, 11	eqsstrdi 3983	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} ⊆ On)
13	12	adantl 486	. . . . 5 ⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → {𝐴} ⊆ On)
14	8, 13	pm2.61dan 824	. . . 4 ⊢ (Ord 𝐴 → {𝐴} ⊆ On)
15	5, 14	unssd 4147	. . 3 ⊢ (Ord 𝐴 → (𝐴 ∪ {𝐴}) ⊆ On)
16	4, 15	eqsstrid 3977	. 2 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ On)
17		ordon 7764	. . 3 ⊢ Ord On
18	17	a1i 11	. 2 ⊢ (Ord 𝐴 → Ord On)
19		trssord 6366	. 2 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
20	3, 16, 18, 19	syl3anc 1394	1 ⊢ (Ord 𝐴 → Ord suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905   ⊆ wss 3907  ∅c0 4288  {csn 4585  Tr wtr 5211  Ord word 6348  Oncon0 6349  suc csuc 6351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-tr 5212  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6352  df-on 6353  df-suc 6355

This theorem is referenced by:  sucexeloni  7796  ordsuc  7798  ord3  8457  ordeldifsucon  43843  ordeldif1o  43844  ordnexbtwnsuc  43851  ordsssucb  43919  onsucunifi  43954