Theorem ssorduni 2999

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 2993	. . 3 ⊢ Ord On
2		trssord 2971	. . . 4 ⊢ ((Tr ∪A ⋀ ∪A ⊆ On ⋀ Ord On) → Ord ∪A)
3	2	3exp 834	. . 3 ⊢ (Tr ∪A → (∪A ⊆ On → (Ord On → Ord ∪A)))
4	1, 3	mpii 45	. 2 ⊢ (Tr ∪A → (∪A ⊆ On → Ord ∪A))
5		ssel 2066	. . . . . . . . 9 ⊢ (A ⊆ On → (y ∈ A → y ∈ On))
6		eloni 2964	. . . . . . . . . 10 ⊢ (y ∈ On → Ord y)
7		ordtr 2968	. . . . . . . . . 10 ⊢ (Ord y → Tr y)
8		trss 2694	. . . . . . . . . 10 ⊢ (Tr y → (x ∈ y → x ⊆ y))
9	6, 7, 8	3syl 20	. . . . . . . . 9 ⊢ (y ∈ On → (x ∈ y → x ⊆ y))
10	5, 9	syl6 22	. . . . . . . 8 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → x ⊆ y)))
11		anc2r 301	. . . . . . . 8 ⊢ ((y ∈ A → (x ∈ y → x ⊆ y)) → (y ∈ A → (x ∈ y → (x ⊆ y ⋀ y ∈ A))))
12	10, 11	syl 10	. . . . . . 7 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → (x ⊆ y ⋀ y ∈ A))))
13		ssuni 2526	. . . . . . 7 ⊢ ((x ⊆ y ⋀ y ∈ A) → x ⊆ ∪A)
14	12, 13	syl8 24	. . . . . 6 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → x ⊆ ∪A)))
15	14	r19.23adv 1749	. . . . 5 ⊢ (A ⊆ On → (∃y ∈ A x ∈ y → x ⊆ ∪A))
16		eluni2 2511	. . . . 5 ⊢ (x ∈ ∪A ↔ ∃y ∈ A x ∈ y)
17	15, 16	syl5ib 206	. . . 4 ⊢ (A ⊆ On → (x ∈ ∪A → x ⊆ ∪A))
18	17	r19.21aiv 1716	. . 3 ⊢ (A ⊆ On → ∀x ∈ ∪Ax ⊆ ∪A)
19		dftr3 2689	. . 3 ⊢ (Tr ∪A ↔ ∀x ∈ ∪Ax ⊆ ∪A)
20	18, 19	sylibr 200	. 2 ⊢ (A ⊆ On → Tr ∪A)
21		ordelord 2976	. . . . . . . . 9 ⊢ ((Ord y ⋀ x ∈ y) → Ord x)
22	21	ex 373	. . . . . . . 8 ⊢ (Ord y → (x ∈ y → Ord x))
23		visset 1816	. . . . . . . . 9 ⊢ x ∈ V
24	23	elon 2963	. . . . . . . 8 ⊢ (x ∈ On ↔ Ord x)
25	22, 24	syl6ibr 213	. . . . . . 7 ⊢ (Ord y → (x ∈ y → x ∈ On))
26	6, 25	syl 10	. . . . . 6 ⊢ (y ∈ On → (x ∈ y → x ∈ On))
27	5, 26	syl6 22	. . . . 5 ⊢ (A ⊆ On → (y ∈ A → (x ∈ y → x ∈ On)))
28	27	r19.23adv 1749	. . . 4 ⊢ (A ⊆ On → (∃y ∈ A x ∈ y → x ∈ On))
29	28, 16	syl5ib 206	. . 3 ⊢ (A ⊆ On → (x ∈ ∪A → x ∈ On))
30	29	ssrdv 2073	. 2 ⊢ (A ⊆ On → ∪A ⊆ On)
31	4, 20, 30	sylc 68	1 ⊢ (A ⊆ On → Ord ∪A)

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 960  ∀wral 1648  ∃wrex 1649   ⊆ wss 2050  ∪cuni 2507  Tr wtr 2685  Ord word 2953  Oncon0 2954

This theorem is referenced by:  ssonunit 3000  orduni 3003  onsucuni 3091  limuni3 3129  tfrlem8 3924  unialeph 4906

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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