Theorem orduninsuc 7854

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 7851. (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 7796	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		id 23	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3		unieq 4877	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ∪ 𝐴 = ∪ if(𝐴 ∈ On, 𝐴, ∅))
4	2, 3	eqeq12d 2776	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = ∪ 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅)))
5		eqeq1 2764	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
6	5	rexbidv 3186	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
7	6	notbid 321	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
8	4, 7	bibi12d 348	. . . 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 6418	. . . . . 6 ⊢ ∅ ∈ On
10	9	elimel 4551	. . . . 5 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
11	10	onuninsuci 7851	. . . 4 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
12	8, 11	dedth 4540	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13		unon 7842	. . . . . 6 ⊢ ∪ On = On
14	13	eqcomi 2769	. . . . 5 ⊢ On = ∪ On
15		onprc 7792	. . . . . . . 8 ⊢ ¬ On ∈ V
16		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucex 7820	. . . . . . . . 9 ⊢ suc 𝑥 ∈ V
18		eleq1 2848	. . . . . . . . 9 ⊢ (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
19	17, 18	mpbiri 261	. . . . . . . 8 ⊢ (On = suc 𝑥 → On ∈ V)
20	15, 19	mto 200	. . . . . . 7 ⊢ ¬ On = suc 𝑥
21	20	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → ¬ On = suc 𝑥)
22	21	nrex 3090	. . . . 5 ⊢ ¬ ∃𝑥 ∈ On On = suc 𝑥
23	14, 22	2th 267	. . . 4 ⊢ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24		id 23	. . . . . 6 ⊢ (𝐴 = On → 𝐴 = On)
25		unieq 4877	. . . . . 6 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
26	24, 25	eqeq12d 2776	. . . . 5 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ On = ∪ On))
27		eqeq1 2764	. . . . . . 7 ⊢ (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
28	27	rexbidv 3186	. . . . . 6 ⊢ (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
29	28	notbid 321	. . . . 5 ⊢ (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
30	26, 29	bibi12d 348	. . . 4 ⊢ (𝐴 = On → ((𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
31	23, 30	mpbiri 261	. . 3 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
32	12, 31	jaoi 871	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
33	1, 32	sylbi 220	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450  ∅c0 4278  ifcif 4481  ∪ cuni 4866  Ord word 6361  Oncon0 6362  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-suc 6368

This theorem is used by:  ordunisuc2  7855  ordzsl  7856  dflim3  7858  nnsuc  7895  onsupsucismax  44281