Theorem iunord 42747

Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7027, but does not use it directly, since ssorduni 7027 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)

Assertion

Ref	Expression
iunord	⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunord

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtr 5775	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
2	1	ralimi 2981	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝐵)
3		triun 4799	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)
4	2, 3	syl 17	. 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)
5		eliun 4556	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6		nfra1 2970	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 Ord 𝐵
7		nfv 1883	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ On
8		rsp 2958	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → Ord 𝐵))
9		ordelon 5785	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
10	9	ex 449	. . . . . 6 ⊢ (Ord 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
11	8, 10	syl6 35	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On)))
12	6, 7, 11	rexlimd 3055	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ On))
13	5, 12	syl5bi 232	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ On))
14	13	ssrdv 3642	. 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On)
15		ordon 7024	. . 3 ⊢ Ord On
16		trssord 5778	. . . 4 ⊢ ((Tr ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On ∧ Ord On) → Ord ∪ 𝑥 ∈ 𝐴 𝐵)
17	16	3exp 1283	. . 3 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → (Ord On → Ord ∪ 𝑥 ∈ 𝐴 𝐵)))
18	15, 17	mpii 46	. 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → Ord ∪ 𝑥 ∈ 𝐴 𝐵))
19	4, 14, 18	sylc 65	1 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  ∪ ciun 4552  Tr wtr 4785  Ord word 5760  Oncon0 5761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765

This theorem is referenced by:  iunordi  42748