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Theorem iunord 48158
Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7785, but does not use it directly, since ssorduni 7785 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 6386 . . . 4 (Ord 𝐵 → Tr 𝐵)
21ralimi 3079 . . 3 (∀𝑥𝐴 Ord 𝐵 → ∀𝑥𝐴 Tr 𝐵)
3 triun 5282 . . 3 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
42, 3syl 17 . 2 (∀𝑥𝐴 Ord 𝐵 → Tr 𝑥𝐴 𝐵)
5 eliun 5002 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
6 nfra1 3277 . . . . 5 𝑥𝑥𝐴 Ord 𝐵
7 nfv 1909 . . . . 5 𝑥 𝑦 ∈ On
8 rsp 3240 . . . . . 6 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → Ord 𝐵))
9 ordelon 6396 . . . . . . 7 ((Ord 𝐵𝑦𝐵) → 𝑦 ∈ On)
109ex 411 . . . . . 6 (Ord 𝐵 → (𝑦𝐵𝑦 ∈ On))
118, 10syl6 35 . . . . 5 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑦 ∈ On)))
126, 7, 11rexlimd 3259 . . . 4 (∀𝑥𝐴 Ord 𝐵 → (∃𝑥𝐴 𝑦𝐵𝑦 ∈ On))
135, 12biimtrid 241 . . 3 (∀𝑥𝐴 Ord 𝐵 → (𝑦 𝑥𝐴 𝐵𝑦 ∈ On))
1413ssrdv 3986 . 2 (∀𝑥𝐴 Ord 𝐵 𝑥𝐴 𝐵 ⊆ On)
15 ordon 7783 . . 3 Ord On
16 trssord 6389 . . . 4 ((Tr 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ On ∧ Ord On) → Ord 𝑥𝐴 𝐵)
17163exp 1116 . . 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 2098  wral 3057  wrex 3066  wss 3947   ciun 4998  Tr wtr 5267  Ord word 6371  Oncon0 6372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-tr 5268  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-ord 6375  df-on 6376
This theorem is referenced by:  iunordi  48159
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