Theorem ordunisuc2 7840

Description: An ordinal equal to its union contains the successor of each of its members. (Contributed by NM, 1-Feb-2005.)

Assertion

Ref	Expression
ordunisuc2	⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))

Distinct variable group:   𝑥,𝐴

Proof of Theorem ordunisuc2

Step	Hyp	Ref	Expression
1		orduninsuc 7839	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2		ralnex 3088	. . 3 ⊢ (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3		onsuc 7809	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
4		eloni 6367	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ On → Ord suc 𝑥)
5	3, 4	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord suc 𝑥)
6		ordtri3 6394	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
7	5, 6	sylan2 605	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
8	7	con2bid 357	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9		onnbtwn 6454	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ suc 𝑥))
10		imnan 405	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝐴 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ suc 𝑥))
11	9, 10	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐴 → ¬ 𝐴 ∈ suc 𝑥))
12	11	con2d 135	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → ¬ 𝑥 ∈ 𝐴))
13		pm2.21 124	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
14	12, 13	syl6 36	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
15	14	adantl 487	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ∈ suc 𝑥 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
16		ax1w 13	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (suc 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
17	15, 16	jaod 873	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
18		eloni 6367	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → Ord 𝑥)
19		ordtri2or 6458	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
20	18, 19	sylan 592	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
21	20	ancoms 464	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
22	21	orcomd 885	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴))
23	22	adantr 486	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴))
24		ordsssuc2 6451	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ∈ suc 𝑥))
25	24	biimpd 232	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 → 𝐴 ∈ suc 𝑥))
26	25	adantr 486	. . . . . . . . . . 11 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝐴 ⊆ 𝑥 → 𝐴 ∈ suc 𝑥))
27		simpr 490	. . . . . . . . . . 11 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
28	26, 27	orim12d 979	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ((𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
29	23, 28	mpd 16	. . . . . . . . 9 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴))
30	29	ex 418	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
31	17, 30	impbid 215	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
32	8, 31	bitr3d 284	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (¬ 𝐴 = suc 𝑥 ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
33	32	pm5.74da 816	. . . . 5 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))))
34		impexp 456	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
35		simpr 490	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
36		ordelon 6381	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
37	36	ex 418	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
38	37	ancrd 561	. . . . . . . 8 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ 𝐴)))
39	35, 38	impbid2 229	. . . . . . 7 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴))
40	39	imbi1d 344	. . . . . 6 ⊢ (Ord 𝐴 → (((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
41	34, 40	bitr3id 288	. . . . 5 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
42	33, 41	bitrd 282	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
43	42	ralbidv2 3181	. . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
44	2, 43	bitr3id 288	. 2 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
45	1, 44	bitrd 282	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ∪ cuni 4867  Ord word 6356  Oncon0 6357  suc csuc 6359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361  df-suc 6363

This theorem is used by:  dflim4  7844  limsuc2  43882