Theorem orduninsuc 7837

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 7834. (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 7779	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		id 23	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3		unieq 4882	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ∪ 𝐴 = ∪ if(𝐴 ∈ On, 𝐴, ∅))
4	2, 3	eqeq12d 2778	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = ∪ 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅)))
5		eqeq1 2766	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
6	5	rexbidv 3188	. . . . . 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 6416	. . . . . 6 ⊢ ∅ ∈ On
10	9	elimel 4556	. . . . 5 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
11	10	onuninsuci 7834	. . . 4 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
12	8, 11	dedth 4545	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13		unon 7825	. . . . . 6 ⊢ ∪ On = On
14	13	eqcomi 2771	. . . . 5 ⊢ On = ∪ On
15		onprc 7775	. . . . . . . 8 ⊢ ¬ On ∈ V
16		vex 3458	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucex 7803	. . . . . . . . 9 ⊢ suc 𝑥 ∈ V
18		eleq1 2850	. . . . . . . . 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 3092	. . . . 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 4882	. . . . . 6 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
26	24, 25	eqeq12d 2778	. . . . 5 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ On = ∪ On))
27		eqeq1 2766	. . . . . . 7 ⊢ (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
28	27	rexbidv 3188	. . . . . 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 870	. 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 860   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  Vcvv 3454  ∅c0 4285  ifcif 4486  ∪ cuni 4871  Ord word 6359  Oncon0 6360  suc csuc 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364  df-suc 6366

This theorem is used by:  ordunisuc2  7838  ordzsl  7839  dflim3  7841  nnsuc  7878  onsupsucismax  44034