Theorem iunord 50339

Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7778, but does not use it directly, since ssorduni 7778 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 6375	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
2	1	ralimi 3108	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝐵)
3		triun 5237	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)
4	2, 3	syl 18	. 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)
5		eliun 4964	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
6		nfra1 3295	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 Ord 𝐵
7		nfv 1941	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ On
8		rsp 3259	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → Ord 𝐵))
9		ordelon 6385	. . . . . . 7 ⊢ ((Ord 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On)
10	9	ex 417	. . . . . 6 ⊢ (Ord 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On))
11	8, 10	syl6 36	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ On)))
12	6, 7, 11	rexlimd 3278	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ On))
13	5, 12	biimtrid 245	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ On))
14	13	ssrdv 3951	. 2 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On)
15		ordon 7776	. . 3 ⊢ Ord On
16		trssord 6378	. . . 4 ⊢ ((Tr ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On ∧ Ord On) → Ord ∪ 𝑥 ∈ 𝐴 𝐵)
17	16	3exp 1135	. . 3 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → (Ord On → Ord ∪ 𝑥 ∈ 𝐴 𝐵)))
18	15, 17	mpii 47	. 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 → (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ On → Ord ∪ 𝑥 ∈ 𝐴 𝐵))
19	4, 14, 18	sylc 66	1 ⊢ (∀𝑥 ∈ 𝐴 Ord 𝐵 → Ord ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913  ∪ ciun 4960  Tr wtr 5222  Ord word 6360  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by:  iunordi  50340