Theorem onuninsuci 7861

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 7864. (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 6499	. . . . . 6 ⊢ ¬ 𝐴 ∈ 𝐴
3		id 22	. . . . . . . 8 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
4		df-suc 6392	. . . . . . . . . . . 12 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	4	eqeq2i 2748	. . . . . . . . . . 11 ⊢ (𝐴 = suc 𝑥 ↔ 𝐴 = (𝑥 ∪ {𝑥}))
6		unieq 4923	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 ∪ {𝑥}) → ∪ 𝐴 = ∪ (𝑥 ∪ {𝑥}))
7	5, 6	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ (𝑥 ∪ {𝑥}))
8		uniun 4935	. . . . . . . . . . 11 ⊢ ∪ (𝑥 ∪ {𝑥}) = (∪ 𝑥 ∪ ∪ {𝑥})
9		unisnv 4932	. . . . . . . . . . . 12 ⊢ ∪ {𝑥} = 𝑥
10	9	uneq2i 4175	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∪ ∪ {𝑥}) = (∪ 𝑥 ∪ 𝑥)
11	8, 10	eqtri 2763	. . . . . . . . . 10 ⊢ ∪ (𝑥 ∪ {𝑥}) = (∪ 𝑥 ∪ 𝑥)
12	7, 11	eqtrdi 2791	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = (∪ 𝑥 ∪ 𝑥))
13		tron 6409	. . . . . . . . . . . 12 ⊢ Tr On
14		eleq1 2827	. . . . . . . . . . . . 13 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
15	1, 14	mpbii 233	. . . . . . . . . . . 12 ⊢ (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
16		trsuc 6473	. . . . . . . . . . . 12 ⊢ ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
17	13, 15, 16	sylancr 587	. . . . . . . . . . 11 ⊢ (𝐴 = suc 𝑥 → 𝑥 ∈ On)
18		ontr 6495	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Tr 𝑥)
19		df-tr 5266	. . . . . . . . . . . 12 ⊢ (Tr 𝑥 ↔ ∪ 𝑥 ⊆ 𝑥)
20	18, 19	sylib 218	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ∪ 𝑥 ⊆ 𝑥)
21	17, 20	syl 17	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → ∪ 𝑥 ⊆ 𝑥)
22		ssequn1 4196	. . . . . . . . . 10 ⊢ (∪ 𝑥 ⊆ 𝑥 ↔ (∪ 𝑥 ∪ 𝑥) = 𝑥)
23	21, 22	sylib 218	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (∪ 𝑥 ∪ 𝑥) = 𝑥)
24	12, 23	eqtrd 2775	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = 𝑥)
25	3, 24	sylan9eqr 2797	. . . . . . 7 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝐴 = 𝑥)
26		vex 3482	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
27	26	sucid 6468	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
28		eleq2 2828	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ suc 𝑥))
29	27, 28	mpbiri 258	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → 𝑥 ∈ 𝐴)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝑥 ∈ 𝐴)
31	25, 30	eqeltrd 2839	. . . . . 6 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝐴 ∈ 𝐴)
32	2, 31	mto 197	. . . . 5 ⊢ ¬ (𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴)
33	32	imnani 400	. . . 4 ⊢ (𝐴 = suc 𝑥 → ¬ 𝐴 = ∪ 𝐴)
34	33	rexlimivw 3149	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = ∪ 𝐴)
35		onuni 7808	. . . . 5 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
36	1, 35	ax-mp 5	. . . 4 ⊢ ∪ 𝐴 ∈ On
37		onuniorsuc 7857	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))
38	1, 37	ax-mp 5	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)
39	38	ori 861	. . . 4 ⊢ (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)
40		suceq 6452	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴)
41	40	rspceeqv 3645	. . . 4 ⊢ ((∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
42	36, 39, 41	sylancr 587	. . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
43	34, 42	impbii 209	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = ∪ 𝐴)
44	43	con2bii 357	1 ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  ∃wrex 3068   ∪ cun 3961   ⊆ wss 3963  {csn 4631  ∪ cuni 4912  Tr wtr 5265  Oncon0 6386  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392

This theorem is referenced by:  orduninsuc  7864