Theorem ordsuc 4540

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 4477	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		en2lp 4531	. . . . . . . . . 10 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)
3		eleq1 2229	. . . . . . . . . . . . 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 654	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 = 𝐴)
8	7	adantl 275	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ¬ 𝑦 = 𝐴)
9		elelsuc 4387	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ suc 𝐴)
10	9	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ suc 𝐴)
11		ordelss 4357	. . . . . . . . . . . . . 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 3141	. . . . . . . . . . . 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 4381	. . . . . . . . 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 1339	. . . . . . 7 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝐴)
20	19	ancom2s 556	. . . . . 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 1863	. . . 4 ⊢ (Ord suc 𝐴 → ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
23		dftr2 4082	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
24	22, 23	sylibr 133	. . 3 ⊢ (Ord suc 𝐴 → Tr 𝐴)
25		sssucid 4393	. . . 4 ⊢ 𝐴 ⊆ suc 𝐴
26		trssord 4358	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
27	25, 26	mp3an2 1315	. . 3 ⊢ ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
28	24, 27	mpancom 419	. 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 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136   ⊆ wss 3116  Tr wtr 4080  Ord word 4340  suc csuc 4343

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-suc 4349

This theorem is referenced by:  nlimsucg  4543  ordpwsucss  4544