Theorem iunord 42220

Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 6877, but does not use it directly, since ssorduni 6877 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 5654	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
2	1	ralimi 2936	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝐵)
3		triun 4694	. . 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 4460	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6		nfra1 2925	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 Ord 𝐵
7		nfv 1830	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ On
8		rsp 2913	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → Ord 𝐵))
9		ordelon 5664	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
10	9	ex 449	. . . . . 6 ⊢ (Ord 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
11	8, 10	syl6 34	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On)))
12	6, 7, 11	rexlimd 3008	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ On))
13	5, 12	syl5bi 231	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ On))
14	13	ssrdv 3574	. 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On)
15		ordon 6874	. . 3 ⊢ Ord On
16		trssord 5657	. . . 4 ⊢ ((Tr ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On ∧ Ord On) → Ord ∪ 𝑥 ∈ 𝐴 𝐵)
17	16	3exp 1256	. . 3 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → (Ord On → Ord ∪ 𝑥 ∈ 𝐴 𝐵)))
18	15, 17	mpii 45	. 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → Ord ∪ 𝑥 ∈ 𝐴 𝐵))
19	4, 14, 18	sylc 63	1 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∪ ciun 4455  Tr wtr 4680  Ord word 5639  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  iunordi  42221