Theorem ordunisuc2 7865

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 7864	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2		ralnex 3072	. . 3 ⊢ (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3		onsuc 7831	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
4		eloni 6394	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ On → Ord suc 𝑥)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord suc 𝑥)
6		ordtri3 6420	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
7	5, 6	sylan2 593	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
8	7	con2bid 354	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) ↔ ¬ 𝐴 = suc 𝑥))
9		onnbtwn 6478	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ suc 𝑥))
10		imnan 399	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝐴 ∈ suc 𝑥) ↔ ¬ (𝑥 ∈ 𝐴 ∧ 𝐴 ∈ suc 𝑥))
11	9, 10	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ∈ 𝐴 → ¬ 𝐴 ∈ suc 𝑥))
12	11	con2d 134	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → ¬ 𝑥 ∈ 𝐴))
13		pm2.21 123	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
14	12, 13	syl6 35	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (𝐴 ∈ suc 𝑥 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
15	14	adantl 481	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ∈ suc 𝑥 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
16		ax-1 6	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
17	16	a1i 11	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (suc 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
18	15, 17	jaod 860	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
19		eloni 6394	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → Ord 𝑥)
20		ordtri2or 6482	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
21	19, 20	sylan 580	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
22	21	ancoms 458	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
23	22	orcomd 872	. . . . . . . . . . 11 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴))
24	23	adantr 480	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴))
25		ordsssuc2 6475	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ∈ suc 𝑥))
26	25	biimpd 229	. . . . . . . . . . . 12 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 → 𝐴 ∈ suc 𝑥))
27	26	adantr 480	. . . . . . . . . . 11 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝐴 ⊆ 𝑥 → 𝐴 ∈ suc 𝑥))
28		simpr 484	. . . . . . . . . . 11 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))
29	27, 28	orim12d 967	. . . . . . . . . 10 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ((𝐴 ⊆ 𝑥 ∨ 𝑥 ∈ 𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
30	24, 29	mpd 15	. . . . . . . . 9 ⊢ (((Ord 𝐴 ∧ 𝑥 ∈ On) ∧ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴))
31	30	ex 412	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴) → (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
32	18, 31	impbid 212	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
33	8, 32	bitr3d 281	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (¬ 𝐴 = suc 𝑥 ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
34	33	pm5.74da 804	. . . . 5 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))))
35		impexp 450	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
36		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
37		ordelon 6408	. . . . . . . . . 10 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
38	37	ex 412	. . . . . . . . 9 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
39	38	ancrd 551	. . . . . . . 8 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → (𝑥 ∈ On ∧ 𝑥 ∈ 𝐴)))
40	36, 39	impbid2 226	. . . . . . 7 ⊢ (Ord 𝐴 → ((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴))
41	40	imbi1d 341	. . . . . 6 ⊢ (Ord 𝐴 → (((𝑥 ∈ On ∧ 𝑥 ∈ 𝐴) → suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
42	35, 41	bitr3id 285	. . . . 5 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
43	34, 42	bitrd 279	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
44	43	ralbidv2 3174	. . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
45	2, 44	bitr3id 285	. 2 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
46	1, 45	bitrd 279	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∪ cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390

This theorem is referenced by:  dflim4  7869  limsuc2  43053