Theorem ordunisuc2 7666

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 7665	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2		ralnex 3163	. . 3 ⊢ (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3		suceloni 7635	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
4		eloni 6261	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ On → Ord suc 𝑥)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord suc 𝑥)
6		ordtri3 6287	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ Ord suc 𝑥) → (𝐴 = suc 𝑥 ↔ ¬ (𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴)))
7	5, 6	sylan2 592	. . . . . . . 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 6342	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 855	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
19		eloni 6261	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → Ord 𝑥)
20		ordtri2or 6346	. . . . . . . . . . . . . 14 ⊢ ((Ord 𝑥 ∧ Ord 𝐴) → (𝑥 ∈ 𝐴 ∨ 𝐴 ⊆ 𝑥))
21	19, 20	sylan 579	. . . . . . . . . . . . 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 867	. . . . . . . . . . 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 6339	. . . . . . . . . . . . 13 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ∈ suc 𝑥))
26	25	biimpd 228	. . . . . . . . . . . 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 961	. . . . . . . . . 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 211	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
33	8, 32	bitr3d 280	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → (¬ 𝐴 = suc 𝑥 ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
34	33	pm5.74da 800	. . . . 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 6275	. . . . . . . . . 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 225	. . . . . . 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 284	. . . . 5 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
43	34, 42	bitrd 278	. . . 4 ⊢ (Ord 𝐴 → ((𝑥 ∈ On → ¬ 𝐴 = suc 𝑥) ↔ (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
44	43	ralbidv2 3118	. . 3 ⊢ (Ord 𝐴 → (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
45	2, 44	bitr3id 284	. 2 ⊢ (Ord 𝐴 → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
46	1, 45	bitrd 278	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∪ cuni 4836  Ord word 6250  Oncon0 6251  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  dflim4  7670  limsuc2  40782