Theorem ordsuc 4547

Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)

Assertion

Ref	Expression
ordsuc	⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Proof of Theorem ordsuc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordsucim 4484	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		en2lp 4538	. . . . . . . . . 10 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)
3		eleq1 2233	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
4	3	biimpac 296	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴) → 𝐴 ∈ 𝑥)
5	4	anim2i 340	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴)) → (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥))
6	5	expr 373	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)))
7	2, 6	mtoi 659	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 = 𝐴)
8	7	adantl 275	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ¬ 𝑦 = 𝐴)
9		elelsuc 4394	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ suc 𝐴)
10	9	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ suc 𝐴)
11		ordelss 4364	. . . . . . . . . . . . . 14 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ⊆ suc 𝐴)
12	10, 11	sylan2 284	. . . . . . . . . . . . 13 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ suc 𝐴)
13	12	sseld 3146	. . . . . . . . . . . 12 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝑦 ∈ 𝑥 → 𝑦 ∈ suc 𝐴))
14	13	expr 373	. . . . . . . . . . 11 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝑥 → 𝑦 ∈ suc 𝐴)))
15	14	pm2.43d 50	. . . . . . . . . 10 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ suc 𝐴))
16	15	impr 377	. . . . . . . . 9 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ suc 𝐴)
17		elsuci 4388	. . . . . . . . 9 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
18	16, 17	syl 14	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
19	8, 18	ecased 1344	. . . . . . 7 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝐴)
20	19	ancom2s 561	. . . . . 6 ⊢ ((Ord suc 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
21	20	ex 114	. . . . 5 ⊢ (Ord suc 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
22	21	alrimivv 1868	. . . 4 ⊢ (Ord suc 𝐴 → ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
23		dftr2 4089	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
24	22, 23	sylibr 133	. . 3 ⊢ (Ord suc 𝐴 → Tr 𝐴)
25		sssucid 4400	. . . 4 ⊢ 𝐴 ⊆ suc 𝐴
26		trssord 4365	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
27	25, 26	mp3an2 1320	. . 3 ⊢ ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
28	24, 27	mpancom 420	. 2 ⊢ (Ord suc 𝐴 → Ord 𝐴)
29	1, 28	impbii 125	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  ∀wal 1346   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121  Tr wtr 4087  Ord word 4347  suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-suc 4356

This theorem is referenced by:  nlimsucg  4550  ordpwsucss  4551