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Theorem ssonuni 4180
Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
Assertion
Ref Expression
ssonuni (A 𝑉 → (A ⊆ On → A On))

Proof of Theorem ssonuni
StepHypRef Expression
1 ssorduni 4179 . 2 (A ⊆ On → Ord A)
2 uniexg 4141 . . 3 (A 𝑉 A V)
3 elong 4076 . . 3 ( A V → ( A On ↔ Ord A))
42, 3syl 14 . 2 (A 𝑉 → ( A On ↔ Ord A))
51, 4syl5ibr 145 1 (A 𝑉 → (A ⊆ On → A On))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  Vcvv 2551  wss 2911   cuni 3571  Ord word 4065  Oncon0 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  ssonunii  4181  onun2  4182  onuni  4186  iunon  5840
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