Theorem onuninsuc 2356

Description: A limit ordinal is not a successor ordinal.

Hypothesis

Ref	Expression
on.1	⊢ A ∈ On

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem onuninsuc

Step	Hyp	Ref	Expression
1		on.1	. . . . . . . 8 ⊢ A ∈ On
2	1	oneirr 2345	. . . . . . 7 ⊢ ¬ A ∈ A
3		id 9	. . . . . . . . 9 ⊢ (A = ∪A → A = ∪A)
4		df-suc 2205	. . . . . . . . . . . . 13 ⊢ suc x = (x ∪ {x})
5	4	cleq2i 1111	. . . . . . . . . . . 12 ⊢ (A = suc x ↔ A = (x ∪ {x}))
6		unieq 1927	. . . . . . . . . . . 12 ⊢ (A = (x ∪ {x}) → ∪A = ∪(x ∪ {x}))
7	5, 6	sylbi 174	. . . . . . . . . . 11 ⊢ (A = suc x → ∪A = ∪(x ∪ {x}))
8		uniun 1934	. . . . . . . . . . . 12 ⊢ ∪(x ∪ {x}) = (∪x ∪ ∪{x})
9		visset 1350	. . . . . . . . . . . . . 14 ⊢ x ∈ V
10	9	unisn 1932	. . . . . . . . . . . . 13 ⊢ ∪{x} = x
11	10	uneq2i 1608	. . . . . . . . . . . 12 ⊢ (∪x ∪ ∪{x}) = (∪x ∪ x)
12	8, 11	eqtr 1119	. . . . . . . . . . 11 ⊢ ∪(x ∪ {x}) = (∪x ∪ x)
13	7, 12	syl6eq 1140	. . . . . . . . . 10 ⊢ (A = suc x → ∪A = (∪x ∪ x))
14		eleq1 1149	. . . . . . . . . . . . 13 ⊢ (A = suc x → (A ∈ On ↔ suc x ∈ On))
15	1, 14	mpbii 168	. . . . . . . . . . . 12 ⊢ (A = suc x → suc x ∈ On)
16		ordon 2238	. . . . . . . . . . . . . 14 ⊢ Ord On
17		ordtr 2213	. . . . . . . . . . . . . 14 ⊢ (Ord On → Tr On)
18	16, 17	ax-mp 6	. . . . . . . . . . . . 13 ⊢ Tr On
19		trsuc 2308	. . . . . . . . . . . . 13 ⊢ ((Tr On ∧ suc x ∈ On) → x ∈ On)
20	18, 19	mpan 518	. . . . . . . . . . . 12 ⊢ (suc x ∈ On → x ∈ On)
21		eloni 2209	. . . . . . . . . . . . . 14 ⊢ (x ∈ On → Ord x)
22		ordtr 2213	. . . . . . . . . . . . . 14 ⊢ (Ord x → Tr x)
23	21, 22	syl 12	. . . . . . . . . . . . 13 ⊢ (x ∈ On → Tr x)
24		df-tr 2042	. . . . . . . . . . . . 13 ⊢ (Tr x ↔ ∪x ⊆ x)
25	23, 24	sylib 173	. . . . . . . . . . . 12 ⊢ (x ∈ On → ∪x ⊆ x)
26	15, 20, 25	3syl 21	. . . . . . . . . . 11 ⊢ (A = suc x → ∪x ⊆ x)
27		ssequn1 1628	. . . . . . . . . . 11 ⊢ (∪x ⊆ x ↔ (∪x ∪ x) = x)
28	26, 27	sylib 173	. . . . . . . . . 10 ⊢ (A = suc x → (∪x ∪ x) = x)
29	13, 28	eqtrd 1128	. . . . . . . . 9 ⊢ (A = suc x → ∪A = x)
30	3, 29	sylan9eqr 1145	. . . . . . . 8 ⊢ ((A = suc x ∧ A = ∪A) → A = x)
31	9	sucid 2304	. . . . . . . . . 10 ⊢ x ∈ suc x
32		eleq2 1150	. . . . . . . . . 10 ⊢ (A = suc x → (x ∈ A ↔ x ∈ suc x))
33	31, 32	mpbiri 169	. . . . . . . . 9 ⊢ (A = suc x → x ∈ A)
34	33	adantr 306	. . . . . . . 8 ⊢ ((A = suc x ∧ A = ∪A) → x ∈ A)
35	30, 34	eqeltrd 1163	. . . . . . 7 ⊢ ((A = suc x ∧ A = ∪A) → A ∈ A)
36	2, 35	mto 93	. . . . . 6 ⊢ ¬ (A = suc x ∧ A = ∪A)
37		imnan 207	. . . . . 6 ⊢ ((A = suc x → ¬ A = ∪A) ↔ ¬ (A = suc x ∧ A = ∪A))
38	36, 37	mpbir 165	. . . . 5 ⊢ (A = suc x → ¬ A = ∪A)
39	38	a1i 7	. . . 4 ⊢ (x ∈ On → (A = suc x → ¬ A = ∪A))
40	39	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ On A = suc x → ¬ A = ∪A)
41	1	onuniorsuc 2355	. . . . . 6 ⊢ (A = ∪A ∨ A = suc ∪A)
42	41	ori 200	. . . . 5 ⊢ (¬ A = ∪A → A = suc ∪A)
43	1	onss 2347	. . . . . 6 ⊢ A ⊆ On
44	1	elisseti 1355	. . . . . . 7 ⊢ A ∈ V
45	44	onuni 2251	. . . . . 6 ⊢ (A ⊆ On → ∪A ∈ On)
46	43, 45	ax-mp 6	. . . . 5 ⊢ ∪A ∈ On
47	42, 46	jctil 240	. . . 4 ⊢ (¬ A = ∪A → (∪A ∈ On ∧ A = suc ∪A))
48		suceq 2288	. . . . . 6 ⊢ (x = ∪A → suc x = suc ∪A)
49	48	cleq2d 1112	. . . . 5 ⊢ (x = ∪A → (A = suc x ↔ A = suc ∪A))
50	49	rcla4ev 1403	. . . 4 ⊢ ((∪A ∈ On ∧ A = suc ∪A) → ∃x ∈ On A = suc x)
51	47, 50	syl 12	. . 3 ⊢ (¬ A = ∪A → ∃x ∈ On A = suc x)
52	40, 51	impbi 139	. 2 ⊢ (∃x ∈ On A = suc x ↔ ¬ A = ∪A)
53	52	bicon2i 194	1 ⊢ (A = ∪A ↔ ¬ ∃x ∈ On A = suc x)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ∪ cun 1485   ⊆ wss 1487  {csn 1808  ∪cuni 1919  Tr wtr 2041  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  orduninsuc 2365

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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