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Theorem iunord 48906
Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7797, but does not use it directly, since ssorduni 7797 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 6399 . . . 4 (Ord 𝐵 → Tr 𝐵)
21ralimi 3080 . . 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 4999 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
6 nfra1 3281 . . . . 5 𝑥𝑥𝐴 Ord 𝐵
7 nfv 1911 . . . . 5 𝑥 𝑦 ∈ On
8 rsp 3244 . . . . . 6 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → Ord 𝐵))
9 ordelon 6409 . . . . . . 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 3263 . . . 4 (∀𝑥𝐴 Ord 𝐵 → (∃𝑥𝐴 𝑦𝐵𝑦 ∈ On))
135, 12biimtrid 242 . . 3 (∀𝑥𝐴 Ord 𝐵 → (𝑦 𝑥𝐴 𝐵𝑦 ∈ On))
1413ssrdv 4000 . 2 (∀𝑥𝐴 Ord 𝐵 𝑥𝐴 𝐵 ⊆ On)
15 ordon 7795 . . 3 Ord On
16 trssord 6402 . . . 4 ((Tr 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ On ∧ Ord On) → Ord 𝑥𝐴 𝐵)
17163exp 1118 . . 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 2105  wral 3058  wrex 3067  wss 3962   ciun 4995  Tr wtr 5264  Ord word 6384  Oncon0 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389
This theorem is referenced by:  iunordi  48907
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