Theorem ordsuc 4560

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 4497	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		en2lp 4551	. . . . . . . . . 10 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ 𝑥)
3		eleq1 2240	. . . . . . . . . . . . 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 664	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 = 𝐴)
8	7	adantl 277	. . . . . . . 8 ⊢ ((Ord suc 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → ¬ 𝑦 = 𝐴)
9		elelsuc 4407	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ suc 𝐴)
10	9	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ suc 𝐴)
11		ordelss 4377	. . . . . . . . . . . . . 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 3154	. . . . . . . . . . . 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 4401	. . . . . . . . 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 1349	. . . . . . 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 1875	. . . 4 ⊢ (Ord suc 𝐴 → ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
23		dftr2 4101	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
24	22, 23	sylibr 134	. . 3 ⊢ (Ord suc 𝐴 → Tr 𝐴)
25		sssucid 4413	. . . 4 ⊢ 𝐴 ⊆ suc 𝐴
26		trssord 4378	. . . 4 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ suc 𝐴 ∧ Ord suc 𝐴) → Ord 𝐴)
27	25, 26	mp3an2 1325	. . 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 708  ∀wal 1351   = wceq 1353   ∈ wcel 2148   ⊆ wss 3129  Tr wtr 4099  Ord word 4360  suc csuc 4363

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4534

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3809  df-tr 4100  df-iord 4364  df-suc 4369

This theorem is referenced by:  nlimsucg  4563  ordpwsucss  4564