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Theorem iunord 49195
Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7799, but does not use it directly, since ssorduni 7799 does not work when 𝐵 is a proper class. (Contributed by Emmett Weisz, 3-Nov-2019.)
Assertion
Ref Expression
iunord (∀𝑥𝐴 Ord 𝐵 → Ord 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunord
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ordtr 6398 . . . 4 (Ord 𝐵 → Tr 𝐵)
21ralimi 3083 . . 3 (∀𝑥𝐴 Ord 𝐵 → ∀𝑥𝐴 Tr 𝐵)
3 triun 5274 . . 3 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
42, 3syl 17 . 2 (∀𝑥𝐴 Ord 𝐵 → Tr 𝑥𝐴 𝐵)
5 eliun 4995 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
6 nfra1 3284 . . . . 5 𝑥𝑥𝐴 Ord 𝐵
7 nfv 1914 . . . . 5 𝑥 𝑦 ∈ On
8 rsp 3247 . . . . . 6 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → Ord 𝐵))
9 ordelon 6408 . . . . . . 7 ((Ord 𝐵𝑦𝐵) → 𝑦 ∈ On)
109ex 412 . . . . . 6 (Ord 𝐵 → (𝑦𝐵𝑦 ∈ On))
118, 10syl6 35 . . . . 5 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑦 ∈ On)))
126, 7, 11rexlimd 3266 . . . 4 (∀𝑥𝐴 Ord 𝐵 → (∃𝑥𝐴 𝑦𝐵𝑦 ∈ On))
135, 12biimtrid 242 . . 3 (∀𝑥𝐴 Ord 𝐵 → (𝑦 𝑥𝐴 𝐵𝑦 ∈ On))
1413ssrdv 3989 . 2 (∀𝑥𝐴 Ord 𝐵 𝑥𝐴 𝐵 ⊆ On)
15 ordon 7797 . . 3 Ord On
16 trssord 6401 . . . 4 ((Tr 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ On ∧ Ord On) → Ord 𝑥𝐴 𝐵)
17163exp 1120 . . 3 (Tr 𝑥𝐴 𝐵 → ( 𝑥𝐴 𝐵 ⊆ On → (Ord On → Ord 𝑥𝐴 𝐵)))
1815, 17mpii 46 . 2 (Tr 𝑥𝐴 𝐵 → ( 𝑥𝐴 𝐵 ⊆ On → Ord 𝑥𝐴 𝐵))
194, 14, 18sylc 65 1 (∀𝑥𝐴 Ord 𝐵 → Ord 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3061  wrex 3070  wss 3951   ciun 4991  Tr wtr 5259  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  iunordi  49196
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