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Theorem iunord 44611
Description: The indexed union of a collection of ordinal numbers 𝐵(𝑥) is ordinal. This proof is based on the proof of ssorduni 7488, but does not use it directly, since ssorduni 7488 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 6203 . . . 4 (Ord 𝐵 → Tr 𝐵)
21ralimi 3165 . . 3 (∀𝑥𝐴 Ord 𝐵 → ∀𝑥𝐴 Tr 𝐵)
3 triun 5182 . . 3 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
42, 3syl 17 . 2 (∀𝑥𝐴 Ord 𝐵 → Tr 𝑥𝐴 𝐵)
5 eliun 4921 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
6 nfra1 3224 . . . . 5 𝑥𝑥𝐴 Ord 𝐵
7 nfv 1908 . . . . 5 𝑥 𝑦 ∈ On
8 rsp 3210 . . . . . 6 (∀𝑥𝐴 Ord 𝐵 → (𝑥𝐴 → Ord 𝐵))
9 ordelon 6213 . . . . . . 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 3322 . . . 4 (∀𝑥𝐴 Ord 𝐵 → (∃𝑥𝐴 𝑦𝐵𝑦 ∈ On))
135, 12syl5bi 243 . . 3 (∀𝑥𝐴 Ord 𝐵 → (𝑦 𝑥𝐴 𝐵𝑦 ∈ On))
1413ssrdv 3977 . 2 (∀𝑥𝐴 Ord 𝐵 𝑥𝐴 𝐵 ⊆ On)
15 ordon 7486 . . 3 Ord On
16 trssord 6206 . . . 4 ((Tr 𝑥𝐴 𝐵 𝑥𝐴 𝐵 ⊆ On ∧ Ord On) → Ord 𝑥𝐴 𝐵)
17163exp 1113 . . 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 2107  wral 3143  wrex 3144  wss 3940   ciun 4917  Tr wtr 5169  Ord word 6188  Oncon0 6189
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192  df-on 6193
This theorem is referenced by:  iunordi  44612
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