Theorem ordunisuc2 7881

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 7880	. 2 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
2		ralnex 3078	. . 3 ⊢ (∀𝑥 ∈ On ¬ 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
3		onsuc 7847	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
4		eloni 6405	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ On → Ord suc 𝑥)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord suc 𝑥)
6		ordtri3 6431	. . . . . . . . 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 6489	. . . . . . . . . . . . 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 858	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ On) → ((𝐴 ∈ suc 𝑥 ∨ suc 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)))
19		eloni 6405	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → Ord 𝑥)
20		ordtri2or 6493	. . . . . . . . . . . . . 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 870	. . . . . . . . . . 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 6486	. . . . . . . . . . . . 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 965	. . . . . . . . . 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 803	. . . . 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 6419	. . . . . . . . . 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 3180	. . 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 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∪ cuni 4931  Ord word 6394  Oncon0 6395  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-suc 6401

This theorem is referenced by:  dflim4  7885  limsuc2  42998