Theorem onuninsuci 7842

Description: An ordinal is equal to its union if and only if it is not the successor of an ordinal. A closed-form generalization of this result is orduninsuc 7845. (Contributed by NM, 18-Feb-2004.)

Hypothesis

Ref	Expression
onssi.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onuninsuci	⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem onuninsuci

Step	Hyp	Ref	Expression
1		onssi.1	. . . . . . 7 ⊢ 𝐴 ∈ On
2	1	onirri 6479	. . . . . 6 ⊢ ¬ 𝐴 ∈ 𝐴
3		id 23	. . . . . . . 8 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
4		df-suc 6370	. . . . . . . . . . . 12 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	4	eqeq2i 2778	. . . . . . . . . . 11 ⊢ (𝐴 = suc 𝑥 ↔ 𝐴 = (𝑥 ∪ {𝑥}))
6		unieq 4885	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 ∪ {𝑥}) → ∪ 𝐴 = ∪ (𝑥 ∪ {𝑥}))
7	5, 6	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ (𝑥 ∪ {𝑥}))
8		uniun 4897	. . . . . . . . . . 11 ⊢ ∪ (𝑥 ∪ {𝑥}) = (∪ 𝑥 ∪ ∪ {𝑥})
9		unisnv 4894	. . . . . . . . . . . 12 ⊢ ∪ {𝑥} = 𝑥
10	9	uneq2i 4119	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∪ ∪ {𝑥}) = (∪ 𝑥 ∪ 𝑥)
11	8, 10	eqtri 2788	. . . . . . . . . 10 ⊢ ∪ (𝑥 ∪ {𝑥}) = (∪ 𝑥 ∪ 𝑥)
12	7, 11	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = (∪ 𝑥 ∪ 𝑥))
13		tron 6387	. . . . . . . . . . . 12 ⊢ Tr On
14		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
15	1, 14	mpbii 236	. . . . . . . . . . . 12 ⊢ (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
16		trsuc 6454	. . . . . . . . . . . 12 ⊢ ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
17	13, 15, 16	sylancr 599	. . . . . . . . . . 11 ⊢ (𝐴 = suc 𝑥 → 𝑥 ∈ On)
18		ontr 6476	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Tr 𝑥)
19		df-tr 5221	. . . . . . . . . . . 12 ⊢ (Tr 𝑥 ↔ ∪ 𝑥 ⊆ 𝑥)
20	18, 19	sylib 221	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ∪ 𝑥 ⊆ 𝑥)
21	17, 20	syl 18	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → ∪ 𝑥 ⊆ 𝑥)
22		ssequn1 4139	. . . . . . . . . 10 ⊢ (∪ 𝑥 ⊆ 𝑥 ↔ (∪ 𝑥 ∪ 𝑥) = 𝑥)
23	21, 22	sylib 221	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (∪ 𝑥 ∪ 𝑥) = 𝑥)
24	12, 23	eqtrd 2800	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = 𝑥)
25	3, 24	sylan9eqr 2822	. . . . . . 7 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝐴 = 𝑥)
26		vex 3461	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
27	26	sucid 6449	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
28		eleq2 2854	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ suc 𝑥))
29	27, 28	mpbiri 261	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → 𝑥 ∈ 𝐴)
30	29	adantr 486	. . . . . . 7 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝑥 ∈ 𝐴)
31	25, 30	eqeltrd 2865	. . . . . 6 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝐴 ∈ 𝐴)
32	2, 31	mto 200	. . . . 5 ⊢ ¬ (𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴)
33	32	imnani 406	. . . 4 ⊢ (𝐴 = suc 𝑥 → ¬ 𝐴 = ∪ 𝐴)
34	33	rexlimivw 3164	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = ∪ 𝐴)
35		onuni 7793	. . . . 5 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
36	1, 35	ax-mp 5	. . . 4 ⊢ ∪ 𝐴 ∈ On
37		onuniorsuc 7839	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
38	1, 37	ax-mp 5	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)
39	38	ori 875	. . . 4 ⊢ (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)
40		suceq 6433	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴)
41	40	rspceeqv 3606	. . . 4 ⊢ ((∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
42	36, 39, 41	sylancr 599	. . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
43	34, 42	impbii 212	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = ∪ 𝐴)
44	43	con2bii 360	1 ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∪ cun 3904   ⊆ wss 3906  {csn 4591  ∪ cuni 4874  Tr wtr 5220  Oncon0 6364  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368  df-suc 6370

This theorem is used by:  orduninsuc  7845