Theorem ordsuc 4600

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 4537	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		en2lp 4591	. . . . . . . . . 10 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)
3		eleq1 2259	. . . . . . . . . . . . 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 665	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 = 𝐴)
8	7	adantl 277	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ¬ 𝑦 = 𝐴)
9		elelsuc 4445	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ suc 𝐴)
10	9	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ suc 𝐴)
11		ordelss 4415	. . . . . . . . . . . . . 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 3183	. . . . . . . . . . . 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 4439	. . . . . . . . 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 1360	. . . . . . 7 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝐴)
20	19	ancom2s 566	. . . . . 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 1889	. . . 4 ⊢ (Ord suc 𝐴 → ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
23		dftr2 4134	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
24	22, 23	sylibr 134	. . 3 ⊢ (Ord suc 𝐴 → Tr 𝐴)
25		sssucid 4451	. . . 4 ⊢ 𝐴 ⊆ suc 𝐴
26		trssord 4416	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
27	25, 26	mp3an2 1336	. . 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 709  ∀wal 1362   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157  Tr wtr 4132  Ord word 4398  suc csuc 4401

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-uni 3841  df-tr 4133  df-iord 4402  df-suc 4407

This theorem is referenced by:  nlimsucg  4603  ordpwsucss  4604