Theorem ssorduni 2249

Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40.

Assertion

Ref	Expression
ssorduni	⊢ (A ⊆ On → Ord ∪A)

Proof of Theorem ssorduni

Step	Hyp	Ref	Expression
1		ordon 2238	. . 3 ⊢ Ord On
2		trssord 2216	. . . 4 ⊢ ((Tr ∪A ∧ ∪A ⊆ On ∧ Ord On) → Ord ∪A)
3	2	3exp 611	. . 3 ⊢ (Tr ∪A → (∪A ⊆ On → (Ord On → Ord ∪A)))
4	1, 3	mpii 45	. 2 ⊢ (Tr ∪A → (∪A ⊆ On → Ord ∪A))
5		ssel 1502	. . . . . . . . 9 ⊢ (A ⊆ On → (y ∈ A → y ∈ On))
6		eloni 2209	. . . . . . . . . 10 ⊢ (y ∈ On → Ord y)
7		ordtr 2213	. . . . . . . . . 10 ⊢ (Ord y → Tr y)
8		trss 2050	. . . . . . . . . 10 ⊢ (Tr y → (x ∈ y → x ⊆ y))
9	6, 7, 8	3syl 21	. . . . . . . . 9 ⊢ (y ∈ On → (x ∈ y → x ⊆ y))
10	5, 9	syl6 23	. . . . . . . 8 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → x ⊆ y)))
11		anc2r 249	. . . . . . . 8 ⊢ ((y ∈ A → (x ∈ y → x ⊆ y)) → (y ∈ A → (x ∈ y → (x ⊆ y ∧ y ∈ A))))
12	10, 11	syl 12	. . . . . . 7 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → (x ⊆ y ∧ y ∈ A))))
13		ssuni 1937	. . . . . . 7 ⊢ ((x ⊆ y ∧ y ∈ A) → x ⊆ ∪A)
14	12, 13	syl8 25	. . . . . 6 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → x ⊆ ∪A)))
15	14	r19.23adv 1286	. . . . 5 ⊢ (A ⊆ On → (∃y ∈ A x ∈ y → x ⊆ ∪A))
16		eluni2 1923	. . . . 5 ⊢ (x ∈ ∪A ↔ ∃y ∈ A x ∈ y)
17	15, 16	syl5ib 181	. . . 4 ⊢ (A ⊆ On → (x ∈ ∪A → x ⊆ ∪A))
18	17	r19.21aiv 1259	. . 3 ⊢ (A ⊆ On → ∀x ∈ ∪ Ax ⊆ ∪A)
19		dftr3 2045	. . 3 ⊢ (Tr ∪A ↔ ∀x ∈ ∪ Ax ⊆ ∪A)
20	18, 19	sylibr 175	. 2 ⊢ (A ⊆ On → Tr ∪A)
21		ordelord 2221	. . . . . . . . 9 ⊢ ((Ord y ∧ x ∈ y) → Ord x)
22	21	exp 291	. . . . . . . 8 ⊢ (Ord y → (x ∈ y → Ord x))
23		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
24	23	elon 2208	. . . . . . . 8 ⊢ (x ∈ On ↔ Ord x)
25	22, 24	syl6ibr 186	. . . . . . 7 ⊢ (Ord y → (x ∈ y → x ∈ On))
26	6, 25	syl 12	. . . . . 6 ⊢ (y ∈ On → (x ∈ y → x ∈ On))
27	5, 26	syl6 23	. . . . 5 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → x ∈ On)))
28	27	r19.23adv 1286	. . . 4 ⊢ (A ⊆ On → (∃y ∈ A x ∈ y → x ∈ On))
29	28, 16	syl5ib 181	. . 3 ⊢ (A ⊆ On → (x ∈ ∪A → x ∈ On))
30	29	ssrdv 1509	. 2 ⊢ (A ⊆ On → ∪A ⊆ On)
31	4, 20, 30	sylc 62	1 ⊢ (A ⊆ On → Ord ∪A)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wel 803   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∪cuni 1919  Tr wtr 2041  Ord word 2198  Oncon0 2199

This theorem is referenced by:  onunit 2250  uniord 2252  onsucuni 2335

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

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