Theorem onuninsuci 7535

Description: A limit ordinal is not a successor ordinal. (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 6265	. . . . . 6 ⊢ ¬ 𝐴 ∈ 𝐴
3		id 22	. . . . . . . 8 ⊢ (𝐴 = ∪ 𝐴 → 𝐴 = ∪ 𝐴)
4		df-suc 6165	. . . . . . . . . . . 12 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	4	eqeq2i 2811	. . . . . . . . . . 11 ⊢ (𝐴 = suc 𝑥 ↔ 𝐴 = (𝑥 ∪ {𝑥}))
6		unieq 4811	. . . . . . . . . . 11 ⊢ (𝐴 = (𝑥 ∪ {𝑥}) → ∪ 𝐴 = ∪ (𝑥 ∪ {𝑥}))
7	5, 6	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = ∪ (𝑥 ∪ {𝑥}))
8		uniun 4823	. . . . . . . . . . 11 ⊢ ∪ (𝑥 ∪ {𝑥}) = (∪ 𝑥 ∪ ∪ {𝑥})
9		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
10	9	unisn 4820	. . . . . . . . . . . 12 ⊢ ∪ {𝑥} = 𝑥
11	10	uneq2i 4087	. . . . . . . . . . 11 ⊢ (∪ 𝑥 ∪ ∪ {𝑥}) = (∪ 𝑥 ∪ 𝑥)
12	8, 11	eqtri 2821	. . . . . . . . . 10 ⊢ ∪ (𝑥 ∪ {𝑥}) = (∪ 𝑥 ∪ 𝑥)
13	7, 12	eqtrdi 2849	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = (∪ 𝑥 ∪ 𝑥))
14		tron 6182	. . . . . . . . . . . 12 ⊢ Tr On
15		eleq1 2877	. . . . . . . . . . . . 13 ⊢ (𝐴 = suc 𝑥 → (𝐴 ∈ On ↔ suc 𝑥 ∈ On))
16	1, 15	mpbii 236	. . . . . . . . . . . 12 ⊢ (𝐴 = suc 𝑥 → suc 𝑥 ∈ On)
17		trsuc 6243	. . . . . . . . . . . 12 ⊢ ((Tr On ∧ suc 𝑥 ∈ On) → 𝑥 ∈ On)
18	14, 16, 17	sylancr 590	. . . . . . . . . . 11 ⊢ (𝐴 = suc 𝑥 → 𝑥 ∈ On)
19		eloni 6169	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → Ord 𝑥)
20		ordtr 6173	. . . . . . . . . . . . 13 ⊢ (Ord 𝑥 → Tr 𝑥)
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → Tr 𝑥)
22		df-tr 5137	. . . . . . . . . . . 12 ⊢ (Tr 𝑥 ↔ ∪ 𝑥 ⊆ 𝑥)
23	21, 22	sylib 221	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ∪ 𝑥 ⊆ 𝑥)
24	18, 23	syl 17	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → ∪ 𝑥 ⊆ 𝑥)
25		ssequn1 4107	. . . . . . . . . 10 ⊢ (∪ 𝑥 ⊆ 𝑥 ↔ (∪ 𝑥 ∪ 𝑥) = 𝑥)
26	24, 25	sylib 221	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (∪ 𝑥 ∪ 𝑥) = 𝑥)
27	13, 26	eqtrd 2833	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → ∪ 𝐴 = 𝑥)
28	3, 27	sylan9eqr 2855	. . . . . . 7 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝐴 = 𝑥)
29	9	sucid 6238	. . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥
30		eleq2 2878	. . . . . . . . 9 ⊢ (𝐴 = suc 𝑥 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ suc 𝑥))
31	29, 30	mpbiri 261	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → 𝑥 ∈ 𝐴)
32	31	adantr 484	. . . . . . 7 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝑥 ∈ 𝐴)
33	28, 32	eqeltrd 2890	. . . . . 6 ⊢ ((𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴) → 𝐴 ∈ 𝐴)
34	2, 33	mto 200	. . . . 5 ⊢ ¬ (𝐴 = suc 𝑥 ∧ 𝐴 = ∪ 𝐴)
35	34	imnani 404	. . . 4 ⊢ (𝐴 = suc 𝑥 → ¬ 𝐴 = ∪ 𝐴)
36	35	rexlimivw 3241	. . 3 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ 𝐴 = ∪ 𝐴)
37		onuni 7488	. . . . 5 ⊢ (𝐴 ∈ On → ∪ 𝐴 ∈ On)
38	1, 37	ax-mp 5	. . . 4 ⊢ ∪ 𝐴 ∈ On
39	1	onuniorsuci 7534	. . . . 5 ⊢ (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴)
40	39	ori 858	. . . 4 ⊢ (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴)
41		suceq 6224	. . . . 5 ⊢ (𝑥 = ∪ 𝐴 → suc 𝑥 = suc ∪ 𝐴)
42	41	rspceeqv 3586	. . . 4 ⊢ ((∪ 𝐴 ∈ On ∧ 𝐴 = suc ∪ 𝐴) → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
43	38, 40, 42	sylancr 590	. . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 → ∃𝑥 ∈ On 𝐴 = suc 𝑥)
44	36, 43	impbii 212	. 2 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ 𝐴 = ∪ 𝐴)
45	44	con2bii 361	1 ⊢ (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ∪ cun 3879   ⊆ wss 3881  {csn 4525  ∪ cuni 4800  Tr wtr 5136  Ord word 6158  Oncon0 6159  suc csuc 6161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-suc 6165

This theorem is referenced by:  orduninsuc  7538