Theorem orduninsuc 2365

Description: A limit ordinal is not a successor ordinal.

Assertion

Ref	Expression
orduninsuc	⊢ (Ord A → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))

Distinct variable group(s):   x,A

Proof of Theorem orduninsuc

Step	Hyp	Ref	Expression
1		ordeleqon 2241	. 2 ⊢ (Ord A ↔ (A ∈ On ∨ A = On))
2		id 9	. . . . . 6 ⊢ (A = if(A ∈ On, A, ∅) → A = if(A ∈ On, A, ∅))
3		unieq 1927	. . . . . 6 ⊢ (A = if(A ∈ On, A, ∅) → ∪A = ∪if(A ∈ On, A, ∅))
4	2, 3	cleq12d 1115	. . . . 5 ⊢ (A = if(A ∈ On, A, ∅) → (A = ∪A ↔ if(A ∈ On, A, ∅) = ∪if(A ∈ On, A, ∅)))
5		cleq1 1107	. . . . . . 7 ⊢ (A = if(A ∈ On, A, ∅) → (A = suc x ↔ if(A ∈ On, A, ∅) = suc x))
6	5	birexdv 1220	. . . . . 6 ⊢ (A = if(A ∈ On, A, ∅) → (∃x ∈ On A = suc x ↔ ∃x ∈ On if(A ∈ On, A, ∅) = suc x))
7	6	negbid 463	. . . . 5 ⊢ (A = if(A ∈ On, A, ∅) → (¬ ∃x ∈ On A = suc x ↔ ¬ ∃x ∈ On if(A ∈ On, A, ∅) = suc x))
8	4, 7	bibi12d 477	. . . 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 2277	. . . . . 6 ⊢ ∅ ∈ On
10	9	elimel 1793	. . . . 5 ⊢ if(A ∈ On, A, ∅) ∈ On
11	10	onuninsuc 2356	. . . 4 ⊢ (if(A ∈ On, A, ∅) = ∪if(A ∈ On, A, ∅) ↔ ¬ ∃x ∈ On if(A ∈ On, A, ∅) = suc x)
12	8, 11	dedth 1784	. . 3 ⊢ (A ∈ On → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
13		unon 2338	. . . . . 6 ⊢ ∪On = On
14	13	cleqcomi 1105	. . . . 5 ⊢ On = ∪On
15		onprc 2240	. . . . . . . 8 ⊢ ¬ On ∈ V
16		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
17	16	sucex 2303	. . . . . . . . 9 ⊢ suc x ∈ V
18		eleq1 1149	. . . . . . . . 9 ⊢ (On = suc x → (On ∈ V ↔ suc x ∈ V))
19	17, 18	mpbiri 169	. . . . . . . 8 ⊢ (On = suc x → On ∈ V)
20	15, 19	mto 93	. . . . . . 7 ⊢ ¬ On = suc x
21	20	a1i 7	. . . . . 6 ⊢ (x ∈ On → ¬ On = suc x)
22	21	nrex 1270	. . . . 5 ⊢ ¬ ∃x ∈ On On = suc x
23	14, 22	2th 540	. . . 4 ⊢ (On = ∪On ↔ ¬ ∃x ∈ On On = suc x)
24		id 9	. . . . . 6 ⊢ (A = On → A = On)
25		unieq 1927	. . . . . 6 ⊢ (A = On → ∪A = ∪On)
26	24, 25	cleq12d 1115	. . . . 5 ⊢ (A = On → (A = ∪A ↔ On = ∪On))
27		cleq1 1107	. . . . . . 7 ⊢ (A = On → (A = suc x ↔ On = suc x))
28	27	birexdv 1220	. . . . . 6 ⊢ (A = On → (∃x ∈ On A = suc x ↔ ∃x ∈ On On = suc x))
29	28	negbid 463	. . . . 5 ⊢ (A = On → (¬ ∃x ∈ On A = suc x ↔ ¬ ∃x ∈ On On = suc x))
30	26, 29	bibi12d 477	. . . 4 ⊢ (A = On → ((A = ∪A ↔ ¬ ∃x ∈ On A = suc x) ↔ (On = ∪On ↔ ¬ ∃x ∈ On On = suc x)))
31	23, 30	mpbiri 169	. . 3 ⊢ (A = On → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
32	12, 31	jaoi 275	. 2 ⊢ ((A ∈ On ∨ A = On) → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))
33	1, 32	sylbi 174	1 ⊢ (Ord A → (A = ∪A ↔ ¬ ∃x ∈ On A = suc x))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  ∅c0 1707  ifcif 1776  ∪cuni 1919  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  ordzsl 2366  dflim3 2368  nnsuc 2389  tfinds 2401

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-if 1777  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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