Theorem ordsuc 4667

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 4604	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		en2lp 4658	. . . . . . . . . 10 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)
3		eleq1 2294	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
4	3	biimpac 298	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴) → 𝐴 ∈ 𝑥)
5	4	anim2i 342	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴)) → (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥))
6	5	expr 375	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝑦 = 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)))
7	2, 6	mtoi 670	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 = 𝐴)
8	7	adantl 277	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ¬ 𝑦 = 𝐴)
9		elelsuc 4512	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ suc 𝐴)
10	9	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ suc 𝐴)
11		ordelss 4482	. . . . . . . . . . . . . 14 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ⊆ suc 𝐴)
12	10, 11	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ suc 𝐴)
13	12	sseld 3227	. . . . . . . . . . . 12 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝑦 ∈ 𝑥 → 𝑦 ∈ suc 𝐴))
14	13	expr 375	. . . . . . . . . . 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 379	. . . . . . . . 9 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ suc 𝐴)
17		elsuci 4506	. . . . . . . . 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 1386	. . . . . . 7 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝐴)
20	19	ancom2s 568	. . . . . 6 ⊢ ((Ord suc 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
21	20	ex 115	. . . . 5 ⊢ (Ord suc 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
22	21	alrimivv 1923	. . . 4 ⊢ (Ord suc 𝐴 → ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
23		dftr2 4194	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
24	22, 23	sylibr 134	. . 3 ⊢ (Ord suc 𝐴 → Tr 𝐴)
25		sssucid 4518	. . . 4 ⊢ 𝐴 ⊆ suc 𝐴
26		trssord 4483	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
27	25, 26	mp3an2 1362	. . 3 ⊢ ((Tr 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
28	24, 27	mpancom 422	. 2 ⊢ (Ord suc 𝐴 → Ord 𝐴)
29	1, 28	impbii 126	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  ∀wal 1396   = wceq 1398   ∈ wcel 2202   ⊆ wss 3201  Tr wtr 4192  Ord word 4465  suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-iord 4469  df-suc 4474

This theorem is referenced by:  nlimsucg  4670  ordpwsucss  4671