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Theorem iunord 47674
Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7762, but does not use it directly, since ssorduni 7762 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 6375 . . . 4 (Ord 𝐵 → Tr 𝐵)
21ralimi 3083 . . 3 (∀𝑥𝐴 Ord 𝐵 → ∀𝑥𝐴 Tr 𝐵)
3 triun 5279 . . 3 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
42, 3syl 17 . 2 (∀𝑥𝐴 Ord 𝐵 → Tr 𝑥𝐴 𝐵)
5 eliun 5000 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
6 nfra1 3281 . . . . 5 𝑥𝑥𝐴 Ord 𝐵
7 nfv 1917 . . . . 5 𝑥 𝑦 ∈ On
8 rsp 3244 . . . . . 6 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → Ord 𝐵))
9 ordelon 6385 . . . . . . 7 ((Ord 𝐵𝑦𝐵) → 𝑦 ∈ On)
109ex 413 . . . . . 6 (Ord 𝐵 → (𝑦𝐵𝑦 ∈ On))
118, 10syl6 35 . . . . 5 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → (𝑦𝐵𝑦 ∈ On)))
126, 7, 11rexlimd 3263 . . . 4 (∀𝑥𝐴 Ord 𝐵 → (∃𝑥𝐴 𝑦𝐵𝑦 ∈ On))
135, 12biimtrid 241 . . 3 (∀𝑥𝐴 Ord 𝐵 → (𝑦 𝑥𝐴 𝐵𝑦 ∈ On))
1413ssrdv 3987 . 2 (∀𝑥𝐴 Ord 𝐵 𝑥𝐴 𝐵 ⊆ On)
15 ordon 7760 . . 3 Ord On
16 trssord 6378 . . . 4 ((Tr 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ On ∧ Ord On) → Ord 𝑥𝐴 𝐵)
17163exp 1119 . . 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 2106  wral 3061  wrex 3070  wss 3947   ciun 4996  Tr wtr 5264  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 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365
This theorem is referenced by:  iunordi  47675
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