Theorem orduninsuc 7843

Description: An ordinal class is equal to its union if and only if it is not the successor of an ordinal. Closed-form generalization of onuninsuci 7840. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
orduninsuc	⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem orduninsuc

Step	Hyp	Ref	Expression
1		ordeleqon 7781	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		id 22	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3		unieq 4899	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ∪ 𝐴 = ∪ if(𝐴 ∈ On, 𝐴, ∅))
4	2, 3	eqeq12d 2752	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = ∪ 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅)))
5		eqeq1 2740	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
6	5	rexbidv 3165	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
7	6	notbid 318	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
8	4, 7	bibi12d 345	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)))
9		0elon 6412	. . . . . 6 ⊢ ∅ ∈ On
10	9	elimel 4575	. . . . 5 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
11	10	onuninsuci 7840	. . . 4 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
12	8, 11	dedth 4564	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13		unon 7830	. . . . . 6 ⊢ ∪ On = On
14	13	eqcomi 2745	. . . . 5 ⊢ On = ∪ On
15		onprc 7777	. . . . . . . 8 ⊢ ¬ On ∈ V
16		vex 3468	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucex 7805	. . . . . . . . 9 ⊢ suc 𝑥 ∈ V
18		eleq1 2823	. . . . . . . . 9 ⊢ (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
19	17, 18	mpbiri 258	. . . . . . . 8 ⊢ (On = suc 𝑥 → On ∈ V)
20	15, 19	mto 197	. . . . . . 7 ⊢ ¬ On = suc 𝑥
21	20	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → ¬ On = suc 𝑥)
22	21	nrex 3065	. . . . 5 ⊢ ¬ ∃𝑥 ∈ On On = suc 𝑥
23	14, 22	2th 264	. . . 4 ⊢ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24		id 22	. . . . . 6 ⊢ (𝐴 = On → 𝐴 = On)
25		unieq 4899	. . . . . 6 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
26	24, 25	eqeq12d 2752	. . . . 5 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ On = ∪ On))
27		eqeq1 2740	. . . . . . 7 ⊢ (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
28	27	rexbidv 3165	. . . . . 6 ⊢ (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
29	28	notbid 318	. . . . 5 ⊢ (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
30	26, 29	bibi12d 345	. . . 4 ⊢ (𝐴 = On → ((𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
31	23, 30	mpbiri 258	. . 3 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
32	12, 31	jaoi 857	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
33	1, 32	sylbi 217	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464  ∅c0 4313  ifcif 4505  ∪ cuni 4888  Ord word 6356  Oncon0 6357  suc csuc 6359

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361  df-suc 6363

This theorem is referenced by:  ordunisuc2  7844  ordzsl  7845  dflim3  7847  nnsuc  7884  onsupsucismax  43278